Steven Strogatz walks us through the first mathematical proof Albert Einstein did when he was a boy: a proof of the Pythagorean theorem.
Einstein, unfortunately, left no such record of his childhood proof. In his Saturday Review essay, he described it in general terms, mentioning only that it relied on "the similarity of triangles." The consensus among Einstein's biographers is that he probably discovered, on his own, a standard textbook proof in which similar triangles (meaning triangles that are like photographic reductions or enlargements of one another) do indeed play a starring role. Walter Isaacson, Jeremy Bernstein, and Banesh Hoffman all come to this deflating conclusion, and each of them describes the steps that Einstein would have followed as he unwittingly reinvented a well-known proof.
Twenty-four years ago, however, an alternative contender for the lost proof emerged. In his book "Fractals, Chaos, Power Laws," the physicist Manfred Schroeder presented a breathtakingly simple proof of the Pythagorean theorem whose provenance he traced to Einstein.
Of course, that breathtaking simplicity later became a hallmark of Einstein's work in physics. See also this brilliant visualization of the Pythagorean theorem
P.S. I love that two of the top three most popular articles on the New Yorker's web site right now are about Albert Einstein.
I'm dreading it. No hope of solving any equations that day, what with the pie-eating contests, the bickering over the merits of pi versus tau (pi times two), and the throwdowns over who can recite more digits of pi. Just stay off the streets at 9:26:53, when the time will approximate pi to ten places: 3.141592653.
The New Yorker's Steven Strogatz on why pi matters.
As Pi Day approaches, it time for a refresher course, courtesy of Steven Strogatz, on what pi actually means and how you can visualize calculating it. It's all about rearranging the pieces of a circle in a calculus-ish sort of way:
Mathematician Steven Strogatz is doing what sounds like a fascinating series of posts on mathematics for adults. From the initial post:
I'll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who'd like to have a second chance at the subject -- but this time from an adult perspective. It's not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it's so enthralling to those who get it.
More subject blogs like this, please. There are lots of art, politics, technology, fashion, economics, typography, photography, and physics blogs out there, but almost none of them appeal to the beginner or interested non-expert. (thx, steve)
This should provide a sufficient amount of "whoa" for the day: mathematically speaking, how are elephants and big cities the same? A: both cities and elephants have developed a similar level of efficiency in the distribution of resources and transportation.
Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you'd expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes -- precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.