Golden spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. A golden spiral with initial radius 1 has the following polar equation:

The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:

or

with e being the base of natural logarithms, a being the initial radius of the spiral, and b such that when θ is a right angle (a quarter turn in either direction):

Therefore, b is given by

The numerical value of b depends on whether the right angle is measured as 90 degrees or as ${\displaystyle \textstyle {\frac {\pi }{2}}}$ radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of ${\displaystyle b}$ (that is, b can also be the negative of this value):

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