Your Kitchen is so Babylonian
This post is very geeky. It meanders almost aimlessly between the subjects of math, history, and my personal life. Please bear with me. I promise the math will be accessible to even the most severe mathphobe who has at least a vague recollection of algebra.

ball-and-chain and I and our kids have spent much happy time at the home of Ralphie and Mrs. Ralphie. Even though I've been to their house an Avogadro's number of times, it was only about six months ago that I noticed their kitchen tile floor. It looks something like this.

Well, it looks nicer that that, since I've only reproduced a black and white sketch of it, but it's the geometry I'm trying to convey, not the aesthetics. A single tile looks like this.
To some, this would only be a pretty pattern, but I remembered that it was a graphical representation of a proof of the Pythagorean Theorem. So as soon as I noticed the tile, I dashed home, grabbed pencil and paper and convinced myself by jotting down the proof.

Now the Pythagorean Theorem isn't just any statement in mathematics. It was known by the Babylonians at least 3,600 years ago. It was proven by the ancient Greeks. It forms the bedrock of geometry and analytic trigonometry since it is used to calculate the distance between two points. Hundreds of different proofs of the Theorem have been discovered (one by President Garfield!) and Elisha Loomis, an early twentieth century professor, published a collection of 367 proofs. (If I had a wish list, that book would be on it.) Just as any educated well-rounded person should be able to recognize a couple of Shakespeare's sonnets even if her training is in engineering, so I propose that any well-educated person in the languages, or humanities, or arts should know the Pythagorean Theorem and be able to prove it. (In fact, Ralphie tells me that in one of the Planet of the Apes books, one of the human protagonists convinces his ape captors that he is intelligent and civilized by jotting down a proof of the Theorem. I would certainly campaign vigorously for voting rights for any chimp which could do the same.)

So, with your indulgence, allow me to present a simple proof of the Pythagorean Theorem memorialized by the tile on Ralphie's kitchen floor. It is so simple you may remember it and use it later to impress your captors.

First, let's just remind ourselves what the Theorem says. It says that for any right triangle (that's a triangle in which one of the angles is 90 degrees) the lengths of the sides obey a certain relationship.
If the length of side opposite the right angle is c, and the lengths of the other two sides are a and b, the Pythagorean Theorem states that a²+b²=c².

Here's the proof. Let's first take a second look at a tile and label some of the lengths on it.
You can see that the tile is composed of a large square of side c which is cut into four identical right triangles and a smaller square in the center of side b-a. So the area of the larger square is the sum of the areas of the smaller square and the four triangles. Since the area of a right triangle is half the product of the base and height, we have

c² = (b – a)² + 4(ab)/2

Simplifying (just squaring b-a) we get

c² = b² – 2ab + a² + 2ab
c² = b² + a²

Tah da! Strangely satisfying, no?

You can learn much more about the Pythagorean Theorem and much less about Ralphie's house here. - served up by Doctor Bean @ 3:24 AM