MONOCHROMATIC AXONOMETRIC: Geometric Fibonacci is Golden

Thursday 8 March 2012

Geometric Fibonacci is Golden

A Fibonacci sequence is a sequence of numbers {F_n} satisfying the recurrence relation, for n > 2,

$F_{n} = F_{n-1}+F_{n-2} \,$

with the first two terms F₁ and F₂ both being 1.

A geometric sequence is a sequence of numbers {a_n} satisfying the recurrence relation, for n > 1,

$a_n = r\,a_{n-1}$

with the first term a₁ and r being fixed numbers.

If we set the first two terms of the Fibonacci sequence F₁ = 1 and F₂ = φ, and make it a geometric sequence with r = φ, that is, for n > 1,

$\varphi^n = \varphi^{n-1} + \varphi^{n-2}\,$

then it can be easily proved that, for φ > 0,

$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,$

which is the golden ratio.