This particular screensaver did not just nicely *illustrate* math, it actually *motivated* research:

<A HREF="https://www.tcd.ie/Physics/research/groups/foams/media/gasket.pdf">A Tisket, a Tasket, an Apollonian Gasket</A>, Dana Mackenzie 

> In the spring of 2007 I had the good fortune to spend a semester at
> the Mathematical Sciences Research Institute in Berkeley. Someone had
> installed a screen-saver program on the computer. Of course, it had to
> be mathematical. 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 below). 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! 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.

<IMG SRC="https://ilorentz.org/beenakker/MO/Apollonian.png"/>

<A HREF="https://www.youtube.com/watch?v=aeWnjSROR8U">Here</A> you can watch the screensaver in action, illustrating <A HREF="https://en.wikipedia.org/wiki/Descartes%27_theorem">Descartes' theorem.</A>