The golden spiral is a very special geometric shape; a logarithmic spiral with the golden ratio φ for its growth factor.
The golden ratio is the number:
The golden spiral gets get wider by a factor of φ every quarter turn it makes; the ratio by which it gets wider is always a golden ratio. The image below illustrates how these spirals are self-similar, which means that the shape is repeated—infinitely— when magnified.
There are many spirals in nature—the nautilus’ shell, the arms of spiral galaxies—that appear to be logarithmic with ratios close to the golden ratio. None of them, though, are exact golden spirals; Some are approximations.
Writing an Equation for the Golden Spiral
The easiest way to write an equation for these spirals is with polar coordinates. That way, the equation for a golden spiral with an initial radius of one will be:
A more general formula, where a is the initial radius of the spiral, is the following:
The growth factor b is defined as b = (ln φ) / Θright, where Θright is a right angle. If we’re working with degrees, Θright will be 90, and the absolute value of b will be 0.0053468. If we’re working with radians, Θright will be 0.3063489.