Golden spiral

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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.

The polar equation for a golden spiral is

r = ab^θ

where a is an arbitrary scale factor.

For θ in degrees we have

<math>ab^{90 ^ \circ} = \phi ab^{0 ^ \circ}</math>

giving

<math>b = \phi ^ \frac{ 1 }{ 90 } </math>

which is approximately 1.00536.

Similarly for θ in radians we have

<math>b= \phi ^ \frac{ 2}{ \pi}</math>

or approximately 1.35846.

A golden spiral should not be confused with a Fibonacci spiral, which approximates a golden spiral, but is not a true logarithmic spiral. Every quarter turn a Fibonacci spiral gets wider not by φ, but by a changing factor related to the ratios of consecutive terms in the Fibonacci sequence. The ratios of consecutive terms in the Fibonacci series approach φ, so that the two spirals are very similar in appearance. A golden spiral can also be approximated by a "whirling rectangle diagram," in which the opposite corners of the rectangles are connected by quarter-circles. (See image to the right).

It is commonly believed that nautilus shells get wider in the pattern of a golden spiral, and hence are related to both φ and the Fibonacci series. The truth is that nautilus shells exhibit logarithmic spiral growth, but not necessarily golden spiral growth.