Tangencies

The following set of nine circles requires five colors, if each pair of tangent circles must have distinct colors. It is an open problem, posed by Ringel, whether five or any finite number of colors is always enough for any circle arrangement (having no triple tangencies).

Ok, I admit it, these pages are merely an excuse for me to try out Cinderella, a nice multiplatform Java application by Kortenkamp and Richter-Gebert for animating this sort of construction.

But anyway, there is some interesting math involved in constructing the circles above:

Inversion

Three Tangent Circles

Four Tangent Circles

Circular Angle Bisectors

Steiner's Porism

Apollonian Circles

Many Circles

See also Paul Kunkel's tangent circles page for even more animated constructions.