Such spirals, where the distance from the origin is a constant to the power of the angle, are called equiangular spirals, that is, a line from the origin to any point on the curve always finds (the tangent to) the curve meeting it at the same angle. Coxeter states that:
This true spiral is closely approximated by the artificial spiral formed by circular quadrants inscribed in the successive squares, as in [the figure above]. (But the true spiral cuts the sides of the squares at very small angles, instead of touching them.)
The above is adapted from H S M Coxeter's book Introduction to Geometry, 1961, page 165.]
Ned May has generated some beautiful pictures based on Fibonacci Spirals using
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