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Apollonian gaskets

The Apollonian gasket is a fractal. It begins with three circles. Each pair of circles touch each other at a single point, and the three points of contact are distinct.

There are two circles that just touch all three:

Since we have added new tangent circles, we can now take new combinations of circles three at a time that are mutually touching, and find the two circles that touch all three. We continue this, filling in the empty spaces, and creating the Apollonian gasket:

The circles invite decoration with colors or depth.

There are also interesting relationships between the curvature of the circles. A circle’s curvature gets bigger as the circle gets smaller. If it has radius 1, the curvature is 1. If the radius is 1/3, the curvature is 3. And so on. Here is an Apollonian gasket showing the curvature of each circle. If you start at the 2 on the left, and follow the circles around the 3, the numbers are 2, 6, 14,…. Do you see a pattern?

And here is an example of an Apollonian gasket we’ve made.

If you’re curious about the name, Apollonius of Perga, the great Greek geometer, solved the more general problem of finding tangents for three circles that aren’t necessarily touching. In that case, there can be up to eight tangent circles.