Equiangular Spiral. Rotation

Coxeter wrote about the Equiangular Spiral (Logarithmic Spiral):

"This curve was first recognized by Descartes. Jacob Bernouilli found it so fascinating that he arranged to have it engraved on his tombstone with the inscription

These words (which E.T. Bell translates as "Though changed I shall arise the same") express a remarkable consequence of the way the curve can be shifted along itself by a dilative rotation: any dilatation has the same effect on it as a rotation, and vice versa." (Coxeter, p. 125)

Coxeter said that Steinhaus describes this property as an optical illusion: "If we turn the book about the vertex, the spiral seems to grow larger or smaller. Two spirals having the same constant angle with the radii are congruent." (Steinhaus, p. 132). Steinhaus wrote about the relation between Logarithmic Spirals and pursuing paths.

Using position vectors you can change the spiral.
This is another example of equiangular spiral.

REFERENCES

Coxeter - Introduction to Geometry (John Whiley and sons)
Hugo Steinhaus, Mathematical Snapshots, Dover Publications (third edition, 1999)
We can read some pages of this book in Google Books: Mathematical Snapshots by Hugo Steinhaus.
D'Arcy Thompson - On Growth and Form. (Cambridge University Press)