sdi

I found something neat! Let me share it with you.

Consider this lovely construction:

It's similar to the various solutions I've posted to Napoleon's problem—it's a square made out of six circles—but it's not quite the same since it's a circumscribed square rather than an inscribed square. I think it's beautiful because the construction exhibits so much symmetry.

(Did any of you ever play the game Riven way back when? If not, I recommend it—it's one of a very few I consider edifying—but in any case, this construction is terribly reminiscent of the art of that game.)

Anyway, if that were all, it'd be cute but not terribly exciting. But what's neat about it is that, by reflecting a few points around,...

...poof! you have a regular octagon! This construction takes only nine circles, compared to my previous best, which took ten. The trick, of course, is that I wasn't specifically trying to place this octagon anywhere in particular, and it happens to be more natural to construct √2 (as I did here) than it is to construct 1/√2 (as I did previously).

Still, those reflections aren't very efficient, and it feels like I ought to be able to construct a regular octagon using eight circles somehow...