Changing the circles certainly can change the limit set. For example, in the shrinking middle circle animation (32K), the four large circles remain fixed while the middle circle, originally tangent to all four fixed circles, shrinks toward the center.
While this animation is simple, it does suggest something interesting. The observed motion toward the edge is repeated on smaller and smaller scales as we look more closely at the edge. Perhaps moving some of the inverting circles produces what might be called fractal motion. The next three animations support this idea by making more elaborate the effect seen in shrinking middle circle.
These three animations illustrate this notion of fractal motion.
In all three, the fixed circles have radius 1 and centers
In the one moving circle animation (68K), there is a single moving circle, of constant radius, staying tangent to one or two of the fixed circles.
In the two moving circles animation (32K), there are two moving circles, of varying radius, staying tangent to one or two of the fixed circles and to one another.
The four moving circles animation (27K), there are four moving circles, of varying radius, staying tangent to one or two of the fixed circles and to two of the moving circles.
Return to Introduction
Return to Limit Sets