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 = abθ
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.
[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.