Sperner’s Lemma

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Image: Wikimedia Commons

Draw a triangle and color its vertices red, green, and blue. Then divide it into as many smaller triangles as you like (the smaller triangles must meet edge to edge and vertex to vertex). Now color the vertices of these smaller triangles using the same three colors. You can do this however you like, with one proviso: The vertices that lie on a side of the large triangle must take the color of either of its ends (so, for instance, the point at the bottom center of the triangle above must be colored either green or blue, not red).

No matter how this is done, there will always exist a small triangle with vertices of three colors. In fact, there will always be an odd number of such triangles.

The Magdeburg Hemispheres

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German scientist Otto von Guericke conducted a memorable experiment on May 8, 1654: He connected two hemispheres, sealed their rims together, and drew out the air between them using a pump of his own devising. The resulting vacuum was so strong that 30 horses could not pull them apart.

At the time the experiment was seen as a strike against Aristotle’s dictum that nature abhors a vacuum. It’s repeated today as a dramatic demonstration of the power of atmospheric pressure.

Villarceau Circles

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How many circles can be drawn through an arbitrary point on a torus? Surprisingly, there are four. Two are obvious: One is parallel to the equatorial plane of the torus, and another is perpendicular to that.

The other two are produced by cutting the torus obliquely at a special angle. They’re named after French astronomer Yvon Villarceau, who first described them in 1848.

The Ellsberg Paradox

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Here are two urns. Urn 1 contains 100 balls, 50 white and 50 black. Urn 2 contains 100 balls, colored black and white in an unknown ratio. You must choose an urn and draw one ball from it, betting on the ball’s color. There are four possibilities:

  • Bet B1: You draw a ball from Urn 1 and bet that it’s black.
  • Bet W1: You draw a ball from Urn 1 and bet that it’s white.
  • Bet B2: You draw a ball from Urn 2 and bet that it’s black.
  • Bet W2: You draw a ball from Urn 2 and bet that it’s white.

If you win your bet you’ll get $100.

If you’re like most people, you don’t have a preference between B1 and W1, nor between B2 and W2. But most people prefer B1 to B2 and W1 to W2. That is, they prefer “the devil they know”: They’d rather choose the urn with the measurable risk than the one with unmeasurable risk.

This is surprising. The expected payoff from Urn 1 is $50. The fact that most people favor B1 to B2 implies that they believe that Urn 2 contains fewer black balls than Urn 1. But these people most often also favor W1 to W2, implying that they believe that Urn 2 also contains fewer white balls, a contradiction.

Ellsberg offered this as evidence of “ambiguity aversion,” a preference in general for known risks over unknown risks. Why people exhibit this preference isn’t clear. Perhaps they associate ambiguity with ignorance, incompetence, or deceit, or possibly they judge that Urn 1 would serve them better over a series of repeated draws.

The principle was popularized by RAND Corporation economist Daniel Ellsberg, of Pentagon Papers fame. This example is from Leonard Wapner’s Unexpected Expectations (2012).

A Separate Peace

After 30 years of searching, acoustic ecologist Gordon Hempton thinks he’s found the “quietest square inch in the United States.” It’s marked by a red pebble that he placed on a log at 47°51’57.5″N, 123°52’13.3″W, in a corner of the Hoh Rainforest in Olympic National Park in western Washington state. The area is actually full of sounds, but the sounds are natural — by quietest, Hempton means that this point is subject to less human-made noise pollution than any other spot in the American wilderness.

Hempton hopes to protect the space by creating a law that would prohibit air traffic overhead. “From a quiet place, you can really feel the impact of even a single jet in the sky,” he told the BBC. “It’s the loudest sound going. The cone of noise it drags behind it expands to fill more than 1,000 square miles. We wanted to see if a point of silence could ripple out in the same way.”

His website, One Square Inch, has more information about his campaign. “Unless something is done,” he told Outside Online, “we’ll see the complete extinction of quiet in the U.S. in our lifetime.”

Misc

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Image: Wikimedia Commons
  • ZZ Top’s first album is called ZZ Top’s First Album.
  • Supreme Court justice Byron White was the NFL’s top rusher in 1940.
  • LOVE ME TENDER is an anagram of DENVER OMELET.
  • Every palindromic number with an even number of digits is divisible by 11.
  • “In great attempts it is glorious even to fail.” — Cassius

From English antiquary John Aubrey’s 1696 Miscellanies: “Anno 1670, not far from Cyrencester, was an Apparition; Being demanded, whether a good Spirit or a bad? Returned no answer, but departed with a curious Perfume and a most melodious Twang.”