Geometry Thereoms - The Three-Sided Circle

I saw my first geometrical proof of the Pythagorean Theorem on a poster in the hallway of my college's Math Department. The elegant simplicity of it was striking and memorable as encountering the elements of Pascal's triangle in the whole number powers of eleven, and just as initially surprising and then intuitively obvious. I stood next to the poster, copying the pair of diagrams onto a sheet of paper. That night I got on the Internet and discovered no less than nineteen distinct pictorial proofs of that same theorem: some unique in that they contained no squares, although the theorem refers to the three of them in its most basic formulation. One was attributed to former U.S. president James Garfield. (How many mathematical proofs has your candidate got under his belt?)

The theorem can be succinctly stated in one sentence: the square of the hypothenuse of a right triangle is equal to the sum of the squares of the two remaining sides. It is this equation that gives a carpenter the length of the diagonal of a rectangle. Depending on what kind of carpenter Joseph of Nazareth was, he probably taught it to his son. The Egyptians knew and used at least a corollary of it; Pythagoras studied in Egypt and probably learned there what the Chinese and the Babylonians both learned, apparently independently. The Scarecrow recited this theorem to the Wizard of Oz -- incorrectly -- to demonstrate the alleged reliability of the brains he's had 'all along'. Fermat famously set the mathematics world on fire by claiming to have discovered an elegant extension of the theorem but not filling in the details. Thomas Hobbes recounts being pulled, in his middle age, into the world of mathematical abstraction and hence logic and philosophy by a friend's copy of Euclid's Elements open to that very theorem.

The three-sided circle is an ancient method of obtaining a perfect right angle. (It's a surprisingly difficult thing to do with any accuracy; you can't just divide a round wheel in half twice and try to get it as close as you can.) Measure out twelve lengths of rope and put a mark at each length: you might use twelve feet, twelve yards, or twelve wraps around your waist or forearm. Tie two ends to form a loop with a circumference of twelve lengths. If you hold the loop where you tied it, and one friend grabs the marker four lengths from your hand, and a third friend holds three lengths from your hand in the opposite direction, then you pull tightly on the loop you'll for a right triangle with the right angle at your hand. (The hypothenuse of the right triangle, its longest side, will be between your two friends and measure five lengths.) This is almost certainly the method the Egyptians used to make the right angles of the pyramids and other ancient monuments. It works because five squared (twenty-five) equals three squared plus four squared (nine plus sixteen): one example of the Pythagorean Theorem.

The Pythagorean Theorem is an amazingly concise and powerful piece of mathematical insight and artifice. Compared to the foundations of first-order logic, or Euclid's axioms of plane geometry, it is unrivaled in economy of space and potential applicability. Its applications are myriad, from engineering and construction to the fundamentals of trigonometry, which in turn dictate the nature of wave motion of every sort from sound to visible light to other forms of electromagnetic radiation. Even the voltage across our domestic electricity circuits rises and falls in a manner that Pythagoras himself would probably appreciate: a sine curve that traces out all the possible lengths of one side of a right triangle with a certain hypothenuse.

This theorem has been pressed into clay tablets, written on papyrus scrolls, and etched into the minds of students of mathematics for thousands of years. If there is intelligent life elsewhere in the universe, we can be sure that they are watching their own version of Scarecrow recite the same mathematical truth. And laughing when he gets it wrong.