Einstein’s First Proof

Finding genius in elementary geometry.

Albert Einstein, General Relativity

1/23/2022 1:36:00 PM

From 2015: Finding genius in elementary geometry.

Finding genius in elementary geometry.

The logical link between the squares and triangles comes via the confusing Step 4. Here’s a way to make peace with it. Try it out for the easiest kind of right triangle, an isosceles right triangle, also known as a 45-45-90 triangle, which is formed by cutting a square in half along its diagonal.

As you can see, four copies of the triangle fit neatly inside the square. Or, said the other way around, the triangle occupies exactly a quarter of the square. That means thatNow for the cruncher. We never said how big the square and the isosceles right triangle were. The ratio of their areas is

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Great article! If you're into Einstein, you might be interested in my novel about zapping the most intelligent human beings from the past into the present and getting any unsolved questions about the world answered. :)

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2 , which says that the areas of the squares add up. That’s the Pythagorean theorem. 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. The second insight is about additivity. Why do the squares add up (Step 6)? It’s because the triangles add up (Step 2), and the squares are proportional to the triangles (Step 4). The logical link between the squares and triangles comes via the confusing Step 4. Here’s a way to make peace with it. Try it out for the easiest kind of right triangle, an isosceles right triangle, also known as a 45-45-90 triangle, which is formed by cutting a square in half along its diagonal. As before, erect a square on its hypotenuse. If we draw dashed lines on the diagonals of that newly built square, the picture looks like the folding instructions for an envelope. As you can see, four copies of the triangle fit neatly inside the square. Or, said the other way around, the triangle occupies exactly a quarter of the square. That means that f =1/4, in the notation above. Now for the cruncher. We never said how big the square and the isosceles right triangle were. The ratio of their areas is always one to four, for any such envelope. It’s a property of the envelope’s shape, not its size. That’s the thrust of Step 4. It’s obvious when you think of it like that, no? The same thing works for any right triangle of any shape. It doesn’t have to be isosceles. The triangle always occupies a certain fraction, f , of the square on its hypotenuse, and that fraction stays the same no matter how big or small they both are. To be sure, the numerical value of f depends on the proportions of the triangle; if it’s a long, flat sliver, the square on its hypotenuse will have a lot more than four times its area, and so f will be a lot less than 1/4. But that numerical value is irrelevant. Einstein’s proof shows that f disappears in the end anyway. It enters stage right, in Step 4, and promptly exits stage left, in Step 6. 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. A Rorschach inkblot has reflectional symmetry: its mirror image matches the original. In Step 4 of his proof, Einstein exploited a symmetry known as scaling. Take a right triangle with a square on its hypotenuse and rescale both of them by the same amount, as if on a photocopier. That rescaling changes some of their features (their areas and side lengths) while leaving others intact (their angles, proportions, and area ratio). It’s the constancy of the area ratio that undergirds Step 4. 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. Think of the courage it required to reformulate nearly all of physics from the ground up, even if it meant revising Newton and Maxwell along the way. Both special and general relativity are also profoundly geometrical theories. They conceive of the universe as having a dimension beyond the usual three; that fourth dimension is time. Rather than considering the distance between two points (a measure of space), the special-relativistic counterpart of the Pythagorean theorem considers the interval between two events (a measure of space-time). 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. The style of his Pythagorean proof, elegant and seemingly effortless, also portends something of the later scientist. Einstein draws a single line in Step 1, after which the Pythagorean theorem falls out like a ripe avocado. The same spirit of minimalism characterizes all of Einstein’s adult work. 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. Finally, although the young Einstein made his proof of the Pythagorean theorem look easy, it surely wasn’t. Remember that, in his Saturday Review essay, he says that it required “much effort.” Later in life, this tenacity—what Einstein referred to as his stubbornness—would serve him well. It took him years to come up with general relativity, and he often felt overwhelmed by the abstract mathematics that the theory required. Although he was mathematically powerful, he was not among the world’s best. (“Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein,” one of his contemporaries, the mathematician David Hilbert, remarked.) Many years after his Pythagorean proof, Einstein shared this lesson with another twelve-year-old who was wrestling with mathematics. On January 3, 1943, a junior-high-school student named Barbara Lee Wilson wrote to him for advice. “Most of the girls in my room have heroes which they write fan mail to,” she began. “You + my uncle who is in the Coast Guard are my heroes.” Wilson told Einstein that she was anxious about her performance in math class: “I have to work longer in it than most of my friends. I worry (perhaps too much).” Four days later, Einstein sent her a reply. “Until now I never dreamed to be something like a hero,” he wrote. “But since you have given me the nomination I feel that I am one.” As for Wilson’s academic concerns? “Do not worry about your difficulties in mathematics,” he told her. “I can assure you that mine are still greater.” is a professor of mathematics at Cornell. He is the author, most recently, of “The Joy of x