Einstein’s First Proof
The proof relies on two insights. The first is that a right triangle can be decomposed into two smaller copies of itself (Steps 1 and 3). That’s a peculiarity of right triangles. If you try instead, for example, to decompose an equilateral triangle into two smaller equilateral triangles, you’ll find that you can’t. So Einstein’s proof reveals why the Pythagorean theorem applies only to right triangles: they’re the only kind made up of smaller copies of themselves.
.. What we’re seeing here is a quintessential use of a symmetry argument. In science and math, we say that something is symmetrical if some aspect of it stays the same despite a change. A sphere, for instance, has rotational symmetry; rotate it about its center and its appearance stays the same.
.. Throughout his career, Einstein would continue to deploy symmetry arguments like a scalpel, getting to the hidden heart of things. He opened his revolutionary 1905 paper on the special theory of relativity by noting an asymmetry in the existing theories of electricity and magnetism: “It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.” Those asymmetries, Einstein sensed, must be a clue to something rotten at the core of physics as it was then formulated. In his mind, everything else—space, time, matter, energy—was up for grabs, but not symmetry.
.. In general relativity, where space-time itself becomes warped and curved by the matter and energy within it, the Pythagorean theorem still has a part to play; it morphs into a quantity called the metric, which measures the space-time separation between infinitesimally close events, for which curvature can temporarily be overlooked. In a sense, Einstein continued his love affair with the Pythagorean theorem all his life.
.. Incredibly, in the part of his special-relativity paper where he revolutionized our notions of space and time, he used no math beyond high-school algebra and geometry.