Proof Without Words

A couple of months ago, Google honored one of the most famous problems in mathematics — known as Fermat’s Last Theorem — on the 410th birth anniversary of the man who came up with the theorem: Pierre De Fermat. Here’s the Goodle doodle from August 17th, 2011:

The Fermat’s Last Theorem is really a higher-dimensional extension of the Pythagoras theorem that operates in 2 dimensions. Do you remember the proof (or, a proof) of the Pythagorean theorem? Here’s one:

Take a look at the right angle triangle shown below, the sides are: a=3, b=4 and c=5. Every side of the triangle is expanded to create squares A, B and C. If we slice square A into smaller squares of 1×1, we get 9 small squares. Similarly, by cutting square B we get 16 small squares. In total, we have 9+16=25 small squares — which can fit into square C which needs 25 1×1 squares.

In other words, the area of square C is equal to the combined areas of A and B. Hence, a² + b² = c²! This is a visual proof of the Pythagoras theorem — a proof without words.

And Fermat’s Last Theorem is a projection of this into a higher dimensional space. He hypothesized that it is not possible to do this kind of perfect-fit re-arrangement for objects that have more than two dimensions. For example, instead of making squares by projecting the sides of the triangle, if you make cubes (i.e. 3 dimensions instead of 2) then you can’t cut those smaller cubes (A and B) in any way that can give you enough to fill the larger cube. Dissecting A and B into cubes of 1x1x1 would give us 27+64=91 small cubies, but the larger cube needs 125 to be completely filled.

Hence, an + bn is not equal to cn when n>2. And that’s known as Fermat’s Last Theorem.

Fermat himself left no proof of his conjecture, and it remained one of the most famous unsolved problems in mathematics. It took around 350 years for mathematicians to prove Fermat’s Last Theorem! The proof involves some heavy mathematics, and beyond the scope of this post (as well as my limited mathematical knowledge.)

What intrigues me though is the idea of ‘proof without words’ (like a layman’s proof I explained above.) For Pythagorean theorem there exists numerous such proofs that require not even a single word to explain. Here are few of my favorite examples of such visual proofs:

I think the following two are really splendid and quite elegant:

Click to embiggen – it may take some time to spot the proofs in these patterns. The second proof by dissection above is actually ascribed to Bhaskara, the great Indian mathematician from the 12th century.

***

Image courtesy: The first demonstration (proof by rearrangement) is from the Wikipedia link on Pythagorean Theorem, and the other two (proofs by dissection) are from this paper. There are many more such proofs on this wonderful page.

A 48-minute documentary about the mathematician who solved Fermat’s Last Theorem is available here.

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