More Squares

[sdi]

I've posted about Napoleon's problem before: can you inscribe a square in a given circle using no more than six circles total (e.g. including the given one)? It's tricky, but in fact you can, and there's a bunch of ways to do it. If your given circle has a radius of 1, the resulting square has a side length of √2.

Later on, I found a way of circumscribing a square around the given circle, still using six circles. This square has a side length of 2.

Later still, I found a way of making this strange one, again using six circles. It's got a side length of √6.

Now, I've played around quite a bit and figured that those were the only three sizes to find—I hadn't tried to prove it or anything, just hadn't seen anything else come up for a while. But I was playing around some more today, trying to see if there's a way to make an octagon out eight circles (my previous best was nine), and I was very surprised that there's another six-circle square out there!

Bonkers! That square has a side length of 2√2. Finding it inspired me to write a quick computer program to look for every possible size of square that can be made in only six circles, but unfortunately there are no others. I've caught 'em all.

Since I had the program put together, I went ahead and searched to see if I could make any eight-circle octagons from any of those squares, but nope, it's not possible. (There may yet be eight-circle octagons that don't start from a square, but this strikes me as pretty unlikely.) Regardless, it's amazing to me that there was more to find over ground I've gone over so many times already! It seems to be good practice to periodically double-check things one thinks one understands.