A Tisket, a Tasket, an Apollonian Gasket

Fractals made of circles do funny things to mathematicians.

Dana Mackenzie in American Scientist:

ScreenHunter_08 Dec. 23 10.09 The program drew an endless assortment of fractals of varying shapes and ingenuity. Every couple minutes the screen would go blank and refresh itself with a completely different fractal. I have to confess that I spent a few idle minutes watching the fractals instead of writing.

One day, a new design popped up on the screen (see the figure above). It was different from all the other fractals. It was made up of simple shapes—circles, in fact—and unlike all the other screen-savers, it had numbers! My attention was immediately drawn to the sequence of numbers running along the bottom edge: 1, 4, 9, 16 … They were the perfect squares! The sequence was 1-squared, 2-squared, 3-squared, and so on.

Before I became a full-time writer, I used to be a mathematician. Seeing those numbers awakened the math geek in me. What did they mean? And what did they have to do with the fractal on the screen? Quickly, before the screen-saver image vanished into the ether, I sketched it on my notepad, making a resolution to find out someday.

As it turned out, the picture on the screen was a special case of a more general construction.

More here.