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As a fan of 'visual' proofs, I love the book Visual Complex Analysis by Tristan Needham.

For example, this picture http://en.wikipedia.org/wiki/File:Pythagoras_algebraic2.svg leads quickly to Pythagoras's theorem.

I'd like to know which are your favourite examples of a clever visualization that proves or helps the intuition for a proof.

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  • $\begingroup$ I've always been fond of the topological proof that all non-constant polynomials have a root, because, even though the rigors of proving the basic homotopy stuff is extreme, it "makes sense." $\endgroup$ – Thomas Andrews Feb 10 '12 at 16:06
  • $\begingroup$ You can see my answer in is this theorem true in optimization theory $\endgroup$ – H. Kabayakawa Jun 3 '12 at 22:04
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You'll find many of such "proofs" here. The first one is particularly freaky.

I'm personally a fan of using graphs of functions to show their properties. Edgy, right?

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Roger B. Nelsen's books Proofs Without Words: Exercises in Visual Thinking, Vol I and II are very nice. I particularly like this construction of his.

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Claudi Alsini and Roger Nelson have published a series of excellent books about what some people call "proofs without words." The most recent of these books is called Icons of Mathematics: An Exploration of Twenty Key Images, was published by MAA in 2011 and has both appealing ideas and mathematics.

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