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# Golden mean

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The Golden mean, also called Golden ratio, Golden section, Golden number or Divine proportion, is the number

  φ = (1 + 51/2) / 2 ≈ 1.6180339887...

This ratio is present in growth patterns of many things in nature such as shell patterns and the curve of ferns. The ancient Egyptians and ancient Greeks already knew the number and, because they regarded it as an aesthetically pleasing ratio, used it often when building monuments (eg the Parthenon).

One reason for its attractiveness can be explained with the help of the [Golden rectangle]?, that is, a rectangle whose sides have the ratio of 1 to φ: (In the drawing φ is written as F.)

       |..........aF.......|

       +----------+--------+   -
       |          |        |   .
       |          |        |   .
       |    A     |    B   |   a
       |          |        |   .
       |          |        |   .
       |          |        |   .
       +----------+--------+   -

       |....a.....|.a(F-1).|

If from this rectangle we remove square A with sides of length a, then the remaining rectangle B is again a Golden rectangle. From this we can derive the equation that

  aφ / a = a / a(φ - 1)

and therefore that

  φ = 1 / (φ - 1)

and φ is indeed the unique positive real number that satisfies this equation. Since this is equivalent with φ2 - φ - 1 = 0 it follows that φ is an algebraic number. It can, however also be shown that φ is an irrational number.

φ turns up a fair bit in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.

[To be better edited & integrated in the text, I know there's already bits above, but I'd like to see a more complete list, so please add more proven examples...]
This "aesthetically pleasing ratio" is found in many man-made constructions:

- Stairs
- Buildings
- Font sizes (?)

But more interestingly, it is usually found in natural shapes:

- Leaves length / width
- On faces, it's everywhere! Ratio mouth width / nose width, etc.
- More examples welcome!

The golden mean is sometimes used as a scientific way of describing beauty?.

Also, see [chaos Theory]?, fractal

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Last edited July 31, 2001 10:50 pm (diff)
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