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File: DissectingACircle

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----
My lastest posts can be
found here:
  * ColinsBlog
----
Previous blog posts:
  * DissectingASquare
  * AnOddityInTennis
  * DecisionTreeForTennis
  * DecisionTreesInGames
  * AMatterOfConvention
  * DoYouNourishOrTarnish
  * BinarySearchReconsidered
  * TwoEqualsFour
  * TheLostPropertyOffice
  * TheForgivingUserInterface
  * SettingUpRSS
  * WithdrawingFromHackerNews
----
Additionally, some earlier writings:
  * RandomWritings.
  * ColinsBlog2010
  * ColinsBlog2009
  * ColinsBlog2008
  * ColinsBlog2007
  * ColinsBlogBefore2007
********
!! Dissecting a Circle - 2011/07/26

[[[>50
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.

Enjoy!
]]]
So last time I talked about the three possibilities we have when
we are DissectingASquare 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 LiverpoolMathematicalSociety 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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