Dissecting A Circle

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Dissecting a Circle - 2011/07/26

This has been a long time coming - sorry.

It turns out that this topic is richly internally intertwingled, and getting any kind of narrative through it is tricky. I've redrafted things any number of times, and I'm still not entirely happy.

But the time has come to tell you something, so I've gone with this.


So last time I talked about the three possibilities we have when we are Dissecting A Square into congruent (identical in size and shape - reflections allowed) pieces:

  • Exactly one piece touches (and hence contains) the centre point
  • There's more than one piece and they all touch the centre point
  • Two or more pieces touch the centre point, but some don't.

Now, what about the circle. If we just cut it like a pizza then we get all the pieces touching the centre. No problem there.

What about the other possibilities?

I was first presented with this as a puzzle over dinner after one meeting of the Liverpool Mathematical Society in 2009. I solved it in a minute or so, and everyone was very impressed. As it happens, though, I now know several people who have solved it almost instantly - sub 5 minutes. Most interestingly, we all have the same, very odd "thing" (call it "Factor X") in our backgrounds, and anyone without "Factor X" takes a long time to solve the puzzle. Having worked that out, I'm much more impressed by those without "Factor X" who have solved it - that seems much more difficult. And no, I'm not going to say what "Factor X" is - that would be too much of a hint.

So anyway, I presented my solution, and conversation moved on. To me there was an air of inevitability about the solution, and I didn't think too much more about it. Until I was told, a few weeks later, that a group of school kids had got a different solution.

I was gobsmacked. I felt that the solution I'd found was probably the only way to do it, and to find that there was another was, to say the least, really surprising.

But wait, there was more to come.

The other solution was actually one of an infinite family of solutions. And even more, there were solutions for each number from 2 onwards. And even more than that, the number of solutions at each level (as it were) grew explosively.

So we had a rapidly rapidly growing infinite family of solutions. And then there was mine (or at least, the one I'd found). Surely that was all.

But no, Joel Haddley found yet another solution, and when we worked on that we realised we had another infinite family. This one was "better behaved" in that there was just one for each odd number from 3 onwards, but now we had two infinite families.

And the original sporadic.

A moment of insight then revealed that the so-called "sporadic" was, in fact, a degenerate case of the second infinite family. So we had two infinite families, one "linear," and one that grows explosively as you get further along.

And that's where we are. Joel continues to work on it, and may have a proof that we've got them all. But I wondered, maybe the square problem wasn't so simple after all.

And it isn't.

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