Favourite proofs with a visualization

Question

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.

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." — Thomas Andrews, Feb 10 '12 at 16:06
You can see my answer in is this theorem true in optimization theory — H. Kabayakawa, Jun 03 '12 at 22:04

score 11 · Accepted Answer · edited Apr 13 '17 at 12:58

11

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?

edited Apr 13 '17 at 12:58

Community

1

answered Feb 10 '12 at 16:00

Nikolaj-K

12,249

score 3 · Answer 2 · answered Feb 10 '12 at 16:08

3

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.

answered Feb 10 '12 at 16:08

David Mitra

74,748

score 2 · Answer 3 · answered Feb 10 '12 at 16:08

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.

Favourite proofs with a visualization

3 Answers3