Tuesday, February 24, 2009

A Physicist's Proof of the Pythagorean Theorem

One night last week, as I lay in bed thinking about the challenges of the economy, terrorism, and global climate change, I thought to myself, What this world really needs is another proof of the Pythagorean theorem.

This pleasant diversion helped me to fall asleep for a few nights in a row, at least until I came up with the following approach. I call it a physicist's proof not only because it's less than mathematically rigorous, but also because the idea is to begin with a triangle of zero size and derive its law of growth. This is a more "dynamical" way of understanding the theorem than the traditional dissection methods.

Without further ado, here is the argument:



We label the sides by x and y, so the hypotenuse is some function f(x,y). We assume f(0,0) = 0. Now let y increase by an infinitesimal amount dy; then the small triangle is similar to the large right triangle, so that df/dy = y/f. Separating, we find f^2 = y^2 + C(x). Repeating the argument with the horizontal leg, we find f^2 = x^2 + D(y). Thus C(x) - x^2 = D(y) - y^2 = E, a constant, whence f^2 = y^2 + x^2 + E. But f(0,0) = 0 implies E=0, so f^2 = y^2 + x^2. QED.

I believe this can be made rigorous using (sin q)' = cos q and (cos q)' = -sin q (relations which can be proved without recourse to the Pythagorean theorem!)

Anyway, having found a solution, I poked around the web to compare it to the extant proofs. As I expected, my approach was not new. The Wikipedia entry on the Pythagorean theorem features this same basic argument, as does this page's amazing list. The idea is attributed to Michael Hardy of the University of Toledo, from 1988. Let this post serve to celebrate the idea's 20th Anniversary.

As long as I'm being unoriginal, let me also link to a cool math page from the website of Contra Costa Community College. Yesterday I was on this site by chance and happened to see two of my recent triangle puzzles!

4 comments:

Lee-Ann Groblicka said...

Isnt pythagorases theorem a law of physics, an experimentally verifiable property of space. If you arrange two straight rulers at right angles using say a compass then it gives a formula for the distance between two points on the rulers (c) in terms of the distance of the points to their intersection (a) & (b). c^2=a^2+b^2. All proofs of this "theorem" simply beg the question and rely on other assumed mathematically expressed properties of space that express the metric structure of flat eulidean space

Lee-Ann Groblicka said...

Pythagorases theorem is a law of physics that is experimentally verified and not provable as such. It expresses the metric structure of a flat newtonian space which is accurate localky. Any mathematical "proofs" simply beg the question by assuming this structure somewhere in their proof.. discuss

JasonZimba said...

I think you are expressing quite a radical view here. Most people would say that there exists such a thing as mathematics apart from physics; that is, I think most people would say that the truth of a mathematical theorem doesn't rest on the outcome of physical measurements.

Anonymous said...

These comments are a year old, but rigorous proofs of this theorem are, imo, lacking, most are elementary and do not sufficiently elucidate the idea of area or the implicit grid on which the figures are moved. I think that you could set up the mathematical structure...proving things like
1) square count on a grid is preserved for motion
(which is obviously dependent on our definition of what is a "moved figure" - I've looked at Euclid's proof of the theorem, and I'm stuck at seeing what structure is necessary to prove the proposition - a rotated figure has the same area, or square count on a grid).
But once the structure is set up, then the physical reality of the proof "makes sense" as a special case.

To Groblicka - if the pythagorean theorem were "proven wrong" it would only be a SUGGESTION that it was wrong - mathematicians would be greatly interested in such an instance, for it would contradict their principles use in the proof - that is, if all the necessary conditions held, and they would probably find one or more such conditions which were not really fulfilled properly.