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.
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:
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).
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.”
The product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square.
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.”
203313 × 657624 = 426756 × 313302
George Hitchings, who won the Nobel Prize in medicine in 1988, proposed to his wife by saying “Incidentally, you’re my fiancée now” as they drove to an event.
John Bardeen, who won the prize in physics in both 1956 and 1972, told his fiancée, “You can be married in the church if you want to, but not to me.”
Hemingway, a Nobelist in literature in 1954, said, “I remember after I got that marriage license I went across from the license bureau to a bar for a drink. The bartender said, ‘What will you have, sir?’ And I said, ‘A glass of hemlock.'”
And Wolfgang Pauli won the Nobel in physics in 1945. Of his ex-wife’s remarriage, he said, “Had she taken a bullfighter I would have understood, but an ordinary chemist!”
English meteorologist Lewis Fry Richardson (1881-1953) spent the last 25 years of his life trying to establish a mathematical theory of the causes of war. In the first of two books on this subject, Arms and Insecurity, he works out a model of arms races using differential equations and reaches the conclusion that
![\frac{d\left ( U + V \right )}{dt}=\left ( k - \alpha \right )\left \{ U + V - \left [ U_{0} + V_{0} - \frac{g + h}{k - \alpha } \right ] \right \}](https://s0.wp.com/latex.php?latex=+%5Cfrac%7Bd%5Cleft+%28+U+%2B+V+%5Cright+%29%7D%7Bdt%7D%3D%5Cleft+%28+k+-+%5Calpha++%5Cright+%29%5Cleft+%5C%7B+U+%2B+V+-+%5Cleft+%5B+U_%7B0%7D+%2B+V_%7B0%7D+-+%5Cfrac%7Bg+%2B+h%7D%7Bk+-+%5Calpha+%7D+%5Cright+%5D+%5Cright+%5C%7D+&bg=ffffff&fg=000&s=2&c=20201002)
where:
U and V are the annual defense budgets of two parties to a conflict
k is a positive constant representing the response to threat
α is a positive constant representing the fatigue and expense of keeping up defenses
U0 and V0 represent cooperations between the parties, tentatively assumed to remain constant
and g and h represent the “grievances and ambitions, provisionally regarded as constant,” on each side.
The term in brackets is a constant, so Richardson predicted that plotting d(U + V)/dt against (U + V) would produce a straight line. He tried this out using the defense budgets of the Franco-Russian and Austro-German alliances for 1909-14 and got this:
“The four points lie close to a straight line, closer, indeed, than one might expect,” he writes. “Since I first drew this diagram, which was shown at the British Association in Cambridge in 1938, and printed in Nature of 29 October of that year, I have been incredulous about the marvelously good fit. Yet there is no simple mistake. … The mere regularity of these phenomena shows that foreign politics had then a rather machine-like quality, intermediate between the predictability of the moon and the freedom of an unmarried young man.”
The extrapolated straight line hits the x axis at U + V = £194 pounds sterling. “As love covereth a multitude of sins, so the good will between the opposing alliances would just have covered £194 million of defense expenditures on the part of the four nations concerned. Their actual expenditure in 1909 was £199 millions; and so began an arms race which led to World War I.”
(Lewis F. Richardson, Arms and Insecurity, 1949.)
“In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.” — Galileo
Two envelopes contain unequal sums of money (for simplicity, assume the two amounts are positive integers). The probability distributions are unknown. You choose an envelope at random, open it, and see that it contains x dollars. Now you must predict whether the total in the other envelope is more or less than x.
Since we know nothing about the other envelope, it would seem we have a 50 percent chance of guessing correctly. But, El Camino College mathematician Leonard Wapner writes, “Unexpectedly, there is something you can do, short of opening the other envelope, to give yourself a better than even chance of getting it right.”
Choose a random positive integer, d, by any means at all. (If d = x then choose again until this isn’t the case.) Now if d > x, guess more, and if d < x, guess less. You’ll guess correctly more than 50 percent of the time.
How is this possible? The random number is chosen independently of the envelopes. How can it point in the direction of the unknown y most of the time? “Think of it this way,” writes Wapner. “If d falls between x and y then your prediction (as indicated by d) is guaranteed to be correct. Assume this occurs with probability p. If d falls less than both x and y, then your prediction will be correct only in the event your chosen number x is the larger of the two. There is a 50 percent chance of this. Similarly, if d is greater than both numbers, your prediction will be correct only if your chosen number is the smaller of the two. This occurs with a 50 percent probability as well.”
So, on balance, your overall probability of being correct is
That’s greater than 0.5, so the odds are in favor of your making a correct prediction.
This example is based on a principle identified by Stanford statistician David Blackwell. “It’s unexpected and ironic that an unrelated random variable can be used to predict that which appears to be completely unpredictable.”
(Leonard M. Wapner, Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball, 2012, following David Blackwell, “On the Translation Parameter Problem for Discrete Variables,” Annals of Mathematical Statistics 22:3 [1951], 393–399.)
