Golden ratio: Encyclopedia II - Golden ratio - Mathematical uses

Golden ratio - Mathematical uses

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 ratio:

The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the Fibonacci sequence is divided by its preceding number, it approximates φ. e.g., 987/610 ≈ 1.6180327868852. Alternatingly the approximation to φ is too small and too large, it gets better as the Fibonacci numbers get higher, and:

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

Because φ is the only positive number that satisfies the identity φⁿ = φ^{n − 1} + φ^{n − 2}, any polynomial expression in φ may be decomposed into a linear expression. For example:

From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It has been argued this is the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). It is also the fundamental unit of the algebraic number field and is a Pisot-Vijayaraghavan number.

The golden ratio has interesting properties when used as the base of a numeral system: see golden mean base.

Adapted from the Wikipedia article "Mathematical uses", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki