Golden mean

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The Golden mean, also called Golden ratio, Golden section, Golden number or Divine proportion, usually denoted by the greek letter 'phi', is the number

φ = (1 + 51/2) / 2 ≈ 1.6180339887...,

the unique positive real number with φ2 = φ + 1.

Two quantities are said to be in the Golden ratio, if the ratio of the bigger to the smaller one is the same as the ratio of the smaller one to their difference. Algebraically, this means a/b = b/(a-b), which, after multiplying both sides by (a-b)/b, leads to (a/b)2 - a/b = 1 and hence a/b = φ.

The ancient Egyptians and ancient Greeks already knew the number and, because they regarded it as an aesthetically pleasing ratio, often used it when building monuments (e.g., the Parthenon). It is also sometimes used in modern man-made constructions, such as stairs and buildings, on in paper formats. Recent studies showed that the Golden ratio plays a role in human perception of beauty, as in body shapes and faces.

A possible reason for its supposed attractiveness is shown by the Golden rectangle, which is a rectangle whose sides a and b stand in the Golden ratio:

   |.......... a..........|

   +-------------+--------+   -
   |             |        |   .
   |             |        |   .
   |      B      |   A    |   b
   |             |        |   .
   |             |        |   .
   |             |        |   .
   +-------------+--------+   -

   |......b......|..a-b...|

If from this rectangle we remove square B with sides of length b, then the remaining rectangle A is again a Golden rectangle, since its side ratio is b/(a-b) = a/b = φ. By iterating this construction, one can produce a sequence of progressively smaller Golden rectangles; by drawing a quarter circle into each of the discarded squares, one obtains a figure which closely resembles the Archimedian spiral.

Since φ is defined to be the root of a polynomial equation, it is an algebraic number. It can be shown that φ is an irrational number.

The number φ turns up frequently 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.

The explicit expression for the Fibonacci sequence involves the golden mean.


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