Geometri
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Geometri (Greek γεωμετρία; geo = bumi, metria = ukuran) adalah sebahagian dari matematik yang mengambil berat dengan soalan saiz, bentuk, dan kedudukan relatif dari rajah dan dengan sifat ruang. Geometri ialah salah satu dari sains tertua. Mulaannya sebuah badan dari pengetahuan praktikal yang mengambil berat dengan jarak, luas dan isipadu, pada abad ketiga S.M. geometri telah diletakkan kepada bentuk aksiometik oleh Euclid, dimana hasilnya mengsetkan piawai untuk beberapa abad berikutnya. Astronomi berkhidmat sebagai sumber penting dari masalah geometrik semasa satu dan separuh alaf seterusnya.
Pengenalan koordinat oleh Descartes dan perkembangan sejajar algebra menandakan peringkat baru untuk geometri, sejak rajah geometrik, seperti lengkung datar, kini boleh dipersembahkan secara analitik. Ini memainkan peranan penting dalam kemunculan kalkulus pada abad ke tujuh belas. Tambahan lagi, teori perspektif menunjukkan bahawa terdapat lebih kepada geometri dari hanya sifat metrik dari rajah. Subjek dari geometri telah selanjutnya diperkayakan oleh pembelajaran struktur intrinsik dari objek geometrik yang berasal dengan Euler dan Gauss dan telah membawa kepada penciptaan topologi dan kebezaan geometri.
Sejak penemuan abad kesembilan-belas dari geometri bukan Euclid, konsep dari ruang telah mengalami perubahan yang hebat. Geometri kotemporari menganggap berganda, ruang yang amat lebih abstrak dari ruang Euclid yang lazim, dimana mereka hanya beranggaran menyerupai pada skala kecil. Ruang ini mungkin dikurniai dengan struktur tambahan, membenarkan seorang untuk bertutur tentang jarak. geometri moden mempunyai ikatan kuat berganda dengan fizik, dicontohi oleh ikatan antara geometri Riemann dan relativiti umum. Salah satu dari teori fizikal termuda, teori tali, ialah juga amat geometrik dalam perasa.
Sifat visual dari geometri membuatkan ia mulaannya lebih mudah diperoleh berbanding dari bahagian lain matematik, seperti algebra atau teori nombor. Bagaimanapun, bahasa geometrik juga digunakan dalam konteks bahawa mereka jauh dikeluarkan dari tradisi, tempat asal Euclidnya, contohnya, dalam geometri fraktal, dan khususnya dalam geometri algebra.[1]
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[Sunting] Sejarah geometri
- Rencana utama: Sejarah geometri

Permulaan geometri terawal yang direkodkan boleh dijejak ke Mesopotamia purba, Mesir, dan Lembah Indus dari sekitar 3000 SM. Geometri awal adalah koleksi dari empirikal yang dijumpai yang mengambil berat jarak, sudut, luas, dan isipadu, dimana telah berkembang untuk menemukan sesetengah keperluan praktikal dalam tinjauan, [[pembinaan, astronomi, dan berbagai kraf. Teks terawal yang dikenali pada geometri ialah Papirus Papirus Mesir, dan Papirus Moscow , Batu bersurat tanah liat Babylonia, dan Shulba Sutras India, manakala orang Cina mempunyai karya Mozi, Zhang Heng, dan Sembilan Bab pada Seni Matematik, di tulis oleh Liu Hui.
Elemen Geometri Euclid (c. 300 SME) merupakan salah satu dari teks awal yang terpenting pada geometri, dimana dia persembahkan geometri dalam bentuk aksiomatik yang ideal, dimana hadir dikenali sebagai geometri Euclid. Treatis ialah bukan, seperti yang kadangkala diingatkan, satu ringkasan dari semua ahli matematik Hellenistik yang seumpama mengetahui tentang geometri pada masa itu; berbanding, ia adalah pengenalan elementari kepadanya;[2] Euclid sendiri menulis lapan lagi buku canggih pada geometri. Kami mengetahui dari rujukan lain bahawa Euclid ialah bukan buku teks elementari geometri pertama, tetapi yang lain jatuh pada tidak dalam kegunaan dan telah hilang.[petikan diperlukan]
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In the Middle Ages, Muslim mathematicians contributed to the development of geometry, especially algebraic geometry and geometric algebra. Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin) (836-901) dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám (1048-1131) found geometric solutions to cubic equations, and his extensive studies of the parallel postulate contributed to the development of Non-Euclidian geometry.[petikan diperlukan]
In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other.
Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann, working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems.
As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics. The traditional type of geometry was recognized as that of homogeneous spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same.
[Sunting] What is geometry?
Recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages. The geometric paradigms presented below should be viewed as 'Pictures at an exhibition' of a sort: they do not exhaust the subject of geometry but rather reflect some of its defining themes.
[Sunting] Practical geometry
There is little doubt that geometry originated as a practical science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.
[Sunting] Axiomatic geometry
A method of computing certain inaccessible distances or heights based on similarity of geometric figures and attributed to Thales presaged more abstract approach to geometry taken by Euclid in his Elements, one of the most influential books ever written. Euclid introduced certain axioms or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigour. In the twentieth century, David Hilbert employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry.
[Sunting] Geometric constructions
Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are the compass and straightedge. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found. The approach to geometric problems with geometric or mechanical means is known as synthetic geometry.
[Sunting] Numbers in geometry
Already Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favour of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the seventeenth century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.
[Sunting] Geometry of position
Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three subdisciplines within present day geometry that deal with these and related questions.
A new chapter in Geometria situs was opened by Leonhard Euler, who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape. Topology, which grew out of geometry, but turned into a large independent discipline, does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.
[Sunting] Geometry beyond Euclid
For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[3] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
[Sunting] Symmetry
The theme of symmetry in geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Nonetheless, it was not until the second half of nineteenth century that the unifying role of symmetry in foundations of geometry had been recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.
[Sunting] Modern geometry
Modern geometry is the title of a popular textbook by Dubrovin, Novikov, and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on manifolds and their applications in contemporary theoretical physics. A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry, and Lie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.
[Sunting] Contemporary geometers
Some of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov, and William Thurston. The common feature in their work is the use of smooth manifolds as the basic idea of space; they otherwise have rather different directions and interests. Geometry now is, in large part, the study of structures on manifolds that have a geometric meaning, in the sense of the principle of covariance that lies at the root of general relativity theory in theoretical physics. (See Category:Structures on manifolds for a survey.)
Much of this theory relates to the theory of continuous symmetry, or in other words Lie groups. From the foundational point of view, on manifolds and their geometrical structures, important is the concept of pseudogroup, defined formally by Shiing-shen Chern in pursuing ideas introduced by Élie Cartan. A pseudogroup can play the role of a Lie group of infinite dimension.
[Sunting] Dimension
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Exactly why is something to which research may bring a satisfactory geometric answer.
[Sunting] Contemporary Euclidean geometry
- Rencana utama: Euclidean geometry
The study of traditional Euclidean geometry is by no means dead. It is now typically presented as the geometry of Euclidean spaces of any dimension, and of the Euclidean group of rigid motions. The fundamental formulae of geometry, such as the Pythagorean theorem, can be presented in this way for a general inner product space.
Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.
[Sunting] Algebraic geometry
The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. After a turbulent period of axiomatization, its foundations are in the twenty-first century on a stable basis. Either one studies the 'classical' case where the spaces are complex manifolds that can be described by algebraic equations; or the scheme theory provides a technically sophisticated theory based on general commutative rings.
The geometric style which was traditionally called the Italian school is now known as birational geometry. It has made progress in the fields of threefolds, singularity theory and moduli spaces, as well as recovering and correcting the bulk of the older results. Objects from algebraic geometry are now commonly applied in string theory, as well as diophantine geometry.
Methods of algebraic geometry rely heavily on sheaf theory and other parts of homological algebra. The Hodge conjecture is an open problem that has gradually taken its place as one of the major questions for mathematicians. For practical applications, Gröbner basis theory and real algebraic geometry are major subfields.
[Sunting] Differential geometry
Differential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physics since the suggestion that space is not flat space. Contemporary differential geometry is intrinsic, meaning that space is a manifold and structure is given by a Riemannian metric, or analogue, locally determining a geometry that is variable from point to point.
This approach contrasts with the extrinsic point of view, where curvature means the way a space bends within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry vector bundles. Fundamental to this approach is the connection between curvature and characteristic classes, as exemplified by the generalized Gauss-Bonnet theorem.
[Sunting] Topology and geometry
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.
[Sunting] Axiomatic and open development
The model of Euclid's Elements, a connected development of geometry as an axiomatic system, is in a tension with René Descartes's reduction of geometry to algebra by means of a coordinate system. There were many champions of synthetic geometry, Euclid-style development of projective geometry, in the nineteenth century, Jakob Steiner being a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating in Hilbert's axioms and regarded as of important pedagogic value, most contemporary geometry is a matter of style. Computational synthetic geometry is now a branch of computer algebra.
The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of pure mathematics, which went on in the period c.1900–c.1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on the subgroup concept) arranges theories according to generalization and specialization. For example affine geometry is more general than Euclidean geometry, and more special than projective geometry. The whole theory of classical groups thereby becomes an aspect of geometry. Their invariant theory, at one point in the nineteenth century taken to be the prospective master geometric theory, is just one aspect of the general representation theory of Lie groups. Using finite fields, the classical groups give rise to finite groups, intensively studied in relation to the finite simple groups; and associated finite geometry, which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides.
An example from recent decades is the twistor theory of Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of sheaf theory on complex manifolds. In contrast, the non-commutative geometry of Alain Connes is a conscious use of geometric language to express phenomena of the theory of von Neumann algebras, and to extend geometry into the domain of ring theory where the commutative law of multiplication is not assumed.
Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by Bourbakiste axiomatization trying to complete the work of David Hilbert, is to create winners and losers. The Ausdehnungslehre (calculus of extension) of Hermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physics such as those derived from quaternions. In the shape of general exterior algebra, it became a beneficiary of the Bourbaki presentation of multilinear algebra, and from 1950 onwards has been ubiquitous. In much the same way, Clifford algebra became popular, helped by a 1957 book Geometric Algebra by Emil Artin. The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit into the pure mathematical world post-Principia Mathematica, is yet unwritten. The situation is analogous to the expulsion of infinitesimals from differential calculus. As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: synthetic differential geometry is an approach to infinitesimals from the side of categorical logic, as non-standard analysis is by means of model theory.
[Sunting] Notes
- ↑ Ia amat lazim dalam geometri algebra untuk bertutur tentang geometri dari aneka algebra atas bidang terbatas, kemungkinan besar seorangan. Dari perspektif jahil, objek ini hanyalah set poin terbatas, tetapi dengan memohon imej goemetrik yang berkuasa dan menggunakan teknik geometrik yang berkembang baik, ia tak mustahil untuk mencari struktur dan mendirikan sifat yang membuatkan mereka sedikit sebanyak setanding kepada sfera atau kon yang biasa.
- ↑ Boyer (1991). bab: “Euclid of Alexandria”, tajuk:, muka surat: 104. "Elemen ialah bukan, seperti yang kadangkala diingatkan, satu ringkasan dari semua pengetahuan geometri; ia sebaliknya ialah pengenalan buku teks yang meliputi semua matematik elementari-"
- ↑ Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.
[Sunting] See also
Carilah maklumat lanjut di Geometry melalui projek-projek kembar Wikipedia:
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[Sunting] Lists
- List of basic geometry topics
- List of geometry topics
- List of geometers
- Category:Geometers
- Category:Algebraic geometers
- Category:Differential geometers
- Category:Topologists
- Important publications in geometry
- List of mathematics articles
[Sunting] Related topics
- Interactive geometry software
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Wikisource mempunyai teks asal berkaitan dengan rencana: Flatland |
- Flatland, a book written by Edwin Abbott Abbott about two and three-dimensional space, to understand the concept of four dimensions
- Why 10 dimensions?
[Sunting] External links
- The Math Forum — Geometry
- The Mathematical Atlas — Geometric Areas of Mathematics
- What Is Geometry? at cut-the-knot
- Geometry at cut-the-knot
- Geometry Step by Step from the Land of the Incas by Antonio Gutierrez.
- Islamic Geometry
- Stanford Encyclopedia of Philosophy:
- Online Interactive Geometric Objects by Elmer G. Wiens
- Arabic mathematics : forgotten brilliance?
- The Geometry Junkyard
- Geometry lessons in PowerPoint All lessons introduce mathematical concepts, step by step, with animations of text, points, lines and figures in general. Solution of problems is also given step by step. Colors are used to give hints and clues to follow the concept or the solution of the problems.
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