# Full Circle

The Rakotzbrücke, in Germany’s Kromlauer Park, is designed to describe a perfect circle when it’s married with its reflection in the Rakotzsee.

This is more aesthetic than practical — the bridge has been standing since 1860, and crossing it is now prohibited.

# A One-Sided Score

Conductor and musical lexicographer Nicolas Slonimsky composed a “Möbius Strip Tease” in 1965, while he was teaching at UCLA. The text reads:

Ach! Professor Möbius, glörious Möbius
Ach, we love your topological,
And, ach, so logical strip!
One-sided inside and two-sided outside!
Ach! euphörius, glörius Möbius Strip-Tease!

Slonimsky described the piece as “a unilateral perpetual rondo in a linearly dodecaphonic vertically consonant counterpoint.” The instructions on the score read: “Copy the music for each performer on a strip of 110-b card stock, 68″ by 6″. Give the strip a half twist to turn it into a Möbius strip.” In performance the endless score rotates perpetually around each musician’s head. (That’s Slonimsky above, trying it out with John Cage.)

The score is here if you’d like to try it yourself. Be careful.

# Just Checking

South African statistician J.E. Kerrich’s 1946 textbook An Experimental Introduction to the Theory of Probability has an odd origin: Kerrich happened to be visiting Denmark during the Nazi invasion of 1940, and the Danes agreed to intern him, along with other British citizens, to prevent their being taken to Germany. While in confinement he tossed a coin 10,000 times and recorded the results, and he wrote up his analysis afterward in the book.

For the record, it landed heads 5,067 times.

# The Blythe Intaglios

Flying between Las Vegas and Blythe, Calif., in 1932, pilot George Palmer looked down and got a surprise — a group of enormous figures had been carved into the surface of the Colorado Desert. They had lain there for a thousand years, but they’re so large that no one had noticed them before. (The largest human figure is more than 50 meters long.)

No one knows for certain who created them; altogether there are several dozen figures, most probably representing mythic characters from Yuman cosmology. What else have we been overlooking?

# The Trust Game

University of Iowa economist Joyce Berg devised this test of social expectation. Two players are each given $10. The two are anonymous to one another and may not communicate. The first player, known as the trustor, is given the option to transfer any part of her$10 to the second player, who is known as the trustee. Whatever she sends will be tripled; if she sends $5 the trustee will receive$15. The trustee then has the option to return any portion of what she’s received. The game is played only once, so the two players have no opportunity to communicate through repeated play.

What should they do? If the two trust one another perfectly, then both stand to double their money — the trustor will give all $10 to the trustee, who now has$40. If she returns half of that, then each player has $20. The trouble is that rational players, who seek to maximize their personal gains, won’t behave this way. If the trustor gives the trustee$10, she can just keep all of it, walking away with $40 and leaving the trustor with nothing. Realizing this, the trustor should send nothing at all, keeping at least the$10 she was given. This is the rational expectation.

But in actual experiment, Berg found that fully 30 of 32 trustors sent money, and they sent an average of \$5.16. This is surprising. “From a rational choice perspective,” she wrote, “subjects who sent money must have believed their expected return would be positive; but given the noncooperative prediction, why would they believe this?”

(Joyce Berg, John Dickhaut, and Kevin McCabe, “Trust, Reciprocity, and Social History,” Games and Economic Behavior 10 [1995], 122-142.)

# Extra Credit

The boys in Nikolay Bogdanov-Belsky’s 1895 painting Mental Arithmetic are having a difficult time solving the problem on the board:

$\displaystyle \frac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365}$

As it happens, there’s a simple solution: Both (102 + 112 + 122) and (132 + 142) are equal to 365, so the answer is simply (365 + 365) / 365, or 2. They’ll figure it out.

# A Box Code

In Robert Chambers’ 1906 novel The Tracer of Lost Persons, Mr. Keen copies the figure above from a mysterious photograph. He is trying to help Captain Harren find a young woman with whom he has become obsessed.

“It’s the strangest cipher I ever encountered,” he says at length. “The strangest I ever heard of. I have seen hundreds of ciphers — hundreds — secret codes of the State Department, secret military codes, elaborate Oriental ciphers, symbols used in commercial transactions, symbols used by criminals and every species of malefactor. And every one of them can be solved with time and patience and a little knowledge of the subject. But this … this is too simple.”

The message reveals the name of the young woman whom Captain Harren has been seeking. What is it?

# Hall’s Marriage Theorem

Suppose we have a group of n men and n women. Each of the women can find some subset of the men whom she would be happy to marry. And each of the men would be happy with any woman who will have him. Is it always possible to pair everyone off into happy marriages?

Clearly this won’t work if, for example, two of the women have their hearts set on the same man and won’t be happy with anyone else. In general, for any subset of the women, we need to be sure that they can reconcile their preferences so that each of them finds a mate.

Surprisingly, though, that’s all that’s required. So long as every subset of women can collectively express interest in a group of men at least as numerous as their own, it will always be possible to marry off the whole group into happy couples.

The theorem was proved by English mathematician Philip Hall in 1935. Another application of the same principle: Shuffle an ordinary deck of 52 playing cards and deal it into 13 piles of 4 cards each. Now it’s always possible to assemble a run of 13 cards, ace through king, by drawing one card from each pile.

# Tilt

As early as the 1st century B.C., the Chinese text Zhou Bi Suan Jing reflected the reasoning of the Pythagorean theorem, showing how to find the hypotenuse of the 3-4-5 triangle. Arrange four 3×4 rectangles around a unit square, as shown, producing a 7×7 square. The diagonals of the four rectangles produce a tilted square. Now, the area of the 7×7 square is 49, and the area of one right triangle with legs 3 and 4 is 6. So the area of the tilted square is 49 – (4 × 6), or 25. This shows that the hypotenuse of each of the right triangles is 5.

In Mathematics and the Aesthetic (2007), Nathalie Sinclair writes, “The Chinese diagram … is the same as one given by the twelfth-century Indian scholar Bhaskara, whose one-word injunction Behold! recorded his sense of awe.”

# Two for One

Mountains on Saturn’s moon Titan are named after mountains in Middle-earth, the fictional setting of J.R.R. Tolkien’s fantasy novels.

The highest peak on Titan is Mount Doom (“Doom Mons”), which rises more than a mile above the surrounding plain. Tolkien’s Mount Doom made its first appearance in The Lord of the Rings in 1954.

By coincidence, science fiction writer Stanley G. Weinbaum had already placed a fictional Mount Doom on Titan in his 1935 story Flight on Titan.

So, in honoring Tolkien, the International Astronomical Union also fulfilled Weinbaum’s vision.