Circle Inversion Fractals

Here we describe some of the work done in projects by Barbara Bemis, Colleen Clancy, Tatiana Cogevina, and Elizabeth Evans.

Inversion was introduced by Apollonious of Perga. Born around 262 BC, Apollonius studied under the successors of Euclid and wrote eight books. He introduced inversion in his last book, Plane Loci.

Given a circle C with center (a, b) and radius r, and given a point (x, y), recall the point (x', y') is the inverse of (x, y) across the circle C if

(i) point (x', y') and point (x, y) lie on the same ray from the center (a, b), and
(ii) the distance from the center to point (x, y) multiplied by the distance from the center to point (x', y') is equal to r2.

The points (x, y) and (x', y') are inverses across the circle C. Here are the formulas for the inverse of a point and of a circle.

Contents of this page:

A. Inversion has several properties readily deduced from the definition
B. Associated with inversion in some collections of circles is a limit set, often a fractal. This is analogous to the fractals generated by IFS; generally, circle inversion limit sets can be viewed as nonlinear IFS.
C. Sometimes, the limit set is more complicated if the circles overlap. Here are some examples. If at the point of intersection, the angles between the tangents is a rational multiple of pi, precise algebraic relations hold between the inversions. These have implications for computing the limit sets.
D. In these animations we show how changing one or more of the inverting circles can affect the limit set. Perhaps more clearly than with other animations, here we get a good sense of fractal motion: the same pattern of movement repeated on smaller and smaller scales.
E. Here is Benoit Mandelbrot's method for rapidly generating images of some limit sets.
F. Here we show how to compute the dimension of some limit sets.
G. Here we present a variant of the limit set, forbiding all combinations of inversions that are expansions.
H. Here we use driven IFS to map which combinations of inversions result in expansions when the inverting cirlces overlap.