This video from Veritasium is a nice explanation of the mathematician David Hilbert's paradox of the Grand Hotel, which illustrates that a hotel with an infinite number of rooms can still accommodate new guests even when it's full. Until it can't, that is. See also Steven Strogatz's explanation of Hilbert's infinite hotel and how Georg Cantor's discovery of different types of infinities complicated the hotel's hospitality. (via digg)

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What do swaying bridges, flashing fireflies, clapping audiences, the far side of the Moon, and beating hearts have in common? Their behavior all has something to do with synchronization. In this video, Veritasium explains why and how spontaneous synchronization appears all the time in the physical world.

I was really into the instability of the Millennium Bridge back when it was first opened (and then rapidly closed), so it was great to hear Steven Strogatz's explanation of the bridge's failure.

Oh, and do go play with Nicky Case's firefly visualization to see how synchronization can arise from really simple rules.

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Happy Pi Day! In celebration of this gloriously nerdy event, mathematician Steven Strogatz wrote about how pi was humanity's first glimpse of the power of calculus and an early effort to come to grips with the idea of infinity.

As a ratio, pi has been around since Babylonian times, but it was the Greek geometer Archimedes, some 2,300 years ago, who first showed how to rigorously estimate the value of pi. Among mathematicians of his time, the concept of infinity was taboo; Aristotle had tried to banish it for being too paradoxical and logically treacherous. In Archimedes's hands, however, infinity became a mathematical workhorse.

He used it to discover the area of a circle, the volume of a sphere and many other properties of curved shapes that had stumped the finest mathematicians before him. In each case, he approximated a curved shape by using a large number of tiny straight lines or flat polygons. The resulting approximations were gemlike, faceted objects that yielded fantastic insights into the original shapes, especially when he imagined using infinitely many, infinitesimally small facets in the process.

Here's a video that runs through Archimedes' method for calculating pi:

Strogatz's piece is an excerpt from his forthcoming book, Infinite Powers: How Calculus Reveals the Secrets of the Universe.

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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.

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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.

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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:

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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)

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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.

(thx, john)

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