Golden spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio.

Specifically, a golden spiral gets wider by a factor of φ every quarter-turn it makes, which means it gets wider by a factor of φ4 (about 6.854) every full turn.

Approximate and true Golden Spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio. (A Fibonacci spiral is not shown, but could be constructed from a similar "whirling rectangle diagram", in which the ratios of the rectangles were based on the terms in the Fibonacci series, rather than phi.)

Fibonacci number

In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation:

That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers also denoted as Fn, for n = 0, 1, … , are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418…

Sometimes this sequence is considered to start at F1 = 1, but it is more common to include F0 = 0. The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India.



A tiling with squares whose sides are successive Fibonacci numbers in length.


A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above.