Napoleon's Problem

[sdi]

I've been having quite a lot of fun playing with geometric constructions lately, and I have another one for you all today: a solution to Napoleon's Problem (that is, inscribing a square in a given circle using only a pair of compasses).

It's a famously tricky problem, and to be honest, I came up with this solution by accident while exploring something else. Nonetheless, I suspect this solution—involving six circles—is the simplest one possible, but I haven't managed to prove it yet.

Comments

[violetcabra]

That is fantastically gorgeous! How fascinating, thank you so much for posting this :-)

[sdi]

You're welcome!

I suppose I should thank you as well, given that turning one of your constructions into SVG is what reintroduced me to this whole field of study!

[violetcabra]

You are very gracious!

[sdi]

It's not a very elegant proof, but I wrote a computer program to search for all possible geometric constructions involving six or less circles, and unless there's a bug in the program, six circles is indeed the best one can do.

What's more, my program found a nicer construction than I did:

It has the merit of being more symmetrical than mine was, and producing three squares instead of just two. ;)