The Pythagorean Theorem: What Is It About?

Although twenty five centuries old, the Pythagorean theorem appears vigorous and ubiquitous. A key to the distance formula in Descartes’s method of coordinates, the theorem is implicitly present in all scientific models and engineering computations involving spatial relationships or trigonometry. An invisible companion to the dot-product operation, it is inherent in equations of mathematical physics and continuum mechanics, be those Laplace, NavierStokes, Maxwell or Yang-Mills. Disguised by axiomatic constructions of linear algebra and Riemannian geometry, it animates both quantum physics and Einstein’s gravitation, which are otherwise so hard to reconcile. And of course, a rare day of a statistician or experimenter goes by without Euclid’s nightmare— fitting any three or more points into a straight line, which is accomplished by Gaussian least squares and hence is also reminiscent of Pythagoras. Quite deservedly, therefore, the philosopher’s pants are proudly displayed in middle-school textbooks to represent, perhaps, the only scientific truth circulating among the general public “with proof.” The most popular one (shown in Figure 1) is very convincing indeed. Yet it pictures the whole issue as a cut-and-paste puzzle and leaves us with a feeling of disproportion: one of the most fundamental facts of nature is due to an ingenious tiling trick. The vast majority of other proofs 1 are similar in nature.