Circle Dissection

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Consider a square. There are all sorts of interesting questions about cutting up a square. For example, there is a way of cutting a square into four pieces that can then be reassembled to form an equilateral triangle.

But that's not where I'm going with this.

Technical term: "identical in size and shape" is called "congruent, and you are allowed to turn them over.
There are many ways of cutting up a square, but we want to consider only those that have all the pieces identical in size and shape.

So dissect a square into identical pieces, and ask - where is the centre point? Is it contained within one of the pieces? Is it on a corner? Is it on an edge? Does every piece touch the centre? Or do some touch, and some not touch?

There are technical questions about what we really mean by "cut up", or what's the technical definition of a "dissection." The generally accepted definition is that a "piece" is defined by its "interior," and while that might not help much, there is a technical defined for that, and so on. All this boils down to this:

  • Use a finite number of pieces,
  • The pieces must be "solid,"
  • Don't try to cheat!
    • If it seems dodgy, it probably is.
So here are the challenges. Dissect a square into identical pieces such that:

One of the things that marks the difference between mathematicians and normal people is that for mathematicians it's not enough just to solve the puzzle - they then ask in how many different ways can it be done?

So in how many ways can each of these things be done? Can you characterise them all? In particular, can you describe every solution to each of those questions? There might be more than you initially suspect.

So having done all that, the next thing is - why a square? What about an equilateral triangle?

So, dissect a triangle into identical pieces such that:

Are all these possible? And in how many ways? Here's a hint, they are all possible - see if you can solve them all.

So triangles, check. Squares, check. Nothing really challenging, so perhaps this is easy. But perhaps not.

with the centre point contained within the interior of a piece?

Hmm. Maybe this game isn't trivial. If you can't do it, can you prove that it's impossible? Because that's the other thing that distinguishes mathematicians from normal people: when something seems impossible, they set out to prove that's impossible,and not just that they haven't succeeded yet.

And so to finish ...

It's (surprisingly!) possible to dissect a circle such that some pieces touch the centre point, and some don't. Can you find one?

There's more than one ...

Part of the Farrago of Fragments.
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