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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

giving

which is approximately 1.00536.

Similarly for θ in radians we have

or approximately 1.35846.

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).

Although it is often suggested that the golden spiral occurs repeatedly in nature (e.g. the arms of spiral galaxies or sunflower heads) , this claim is rarely valid except perhaps in the most contrived of circumstances. For example, 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. In truth nautilus shells exhibit logarithmic spiral growth, but not necessarily golden spiral growth. The reason for this growth pattern is that it allows the animal to grow at a constant rate without having to change shape. Spirals are common features in nature, but there is no evidence that a single number dictates the shape of every one of these spirals. The greatest misconception in the mystification of the golden spiral is the incorrect assumption that all spirals in nature are in fact the golden spiral. While logarithmic spirals are often observed, they may be of differing pitches, and therefore there is no single "spira mirabilis".