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Dissecting a Square (Part 2) - 2011/08/08

If you haven't already, you might want to read: Dissecting A Circle Dissecting A Square We're talking about dissections of squares and circles into congruent pieces. That means the pieces are all the same size and shape, although they may be mirror images of each other.

So we return to the square. It's simple enough to cut it up into identical pieces so that all the pieces touch the centre. One way is just to cut it into four identical squares. They all touch the centre. Another way is to cut along just the diagonal, giving two identical (triangular) pieces, both touching the centre.

So it can be done, but in how many ways?

I rapidly got 5 (or 6, depending on a technicality), and I started to wonder about a proof that 5 (or 6) was all of them. I posted a badly worded question on an internet forum, and rightly got flamed for it, but in the answers was a shock.

There was an infinite family of solutions.

In fact there were two infinite families.

No, three!

No, wait, it's not the first or second infinity, it's an infinity beyond that!

Oh crap. I really, really don't know anything at all.

But the infinite families were (and are) quite well behaved, so I started to wonder if it wasn't so bad after all. I mean, I had three easily described infinite families , maybe I could prove that I had them all now!

The families have two, four or eight pieces, all of which divide the number of symmetries of a square. There are clues there, maybe I can work on that.

Then someone produced a "dissection" into 16 pieces.

Oh, crap.

Then someone else produced a dissection into 32 pieces, and wondered openly about 128.

Oh, shoot, now someone in the comments has found another, and I've just generalised it a little to get still another. Now I really really know that I know nothing.

Oh, crapity crap. Once again, I now know that I know nothing.

And there it stands. I haven't told you everything yet, as you might expect, but I hope I've made you at least a little curious.

So here's your puzzle for next time. Dissect a square into identical (size and shape) pieces so that all of them touch the centre point. This can be done in infinitely many ways, but your challenge is to find some and describe them.

Good luck!

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