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.

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.)

A Fibonacci spiral that approximates the Golden Spiral.

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, and b represents the factor by which r increases when θ increases by one degree.

Remembering that r gets larger by a factor of φ when θ increases from 0 to 90 degrees, for θ in degrees we have

$ab^{90 ^ \circ} = \phi ab^{0 ^ \circ}$

giving

$b = \phi ^ \frac{ 1 }{ 90 }$

which is approximately 1.00536.

Similarly for θ in radians we have

$b= \phi ^ \frac{ 2}{ \pi}$

or approximately 1.35846.

[edit] Approximations of the Golden Spiral

There are several similar spirals that approximate, but do not exactly equal, a golden spiral. These are often confused with the golden spiral.

For example, a golden spiral can be approximated by a "whirling rectangle diagram," in which the opposite corners of squares formed by spiraling golden rectangles are connected by quarter-circles. The result is very similar to a true golden spiral (See image on top right).

Another approximation is a Fibonacci spiral, which 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. (See image on bottom 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.

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Categories: Articles lacking sources | Spirals | Golden ratio

Golden spiral

From Wikipedia, the free encyclopedia

[edit] Approximations of the Golden Spiral

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