# Viviani’s Theorem

Pick any point inside an equilateral triangle and measure the distances to its sides.

The sum of those distances is the altitude of the triangle.

# Seeing and Believing

John Dalton was a tornado of English science, exploring atomic theory, meteorology, perception, and the physics of gases with equal avidity.

But he was a Quaker, and when in 1834 he was invited to be presented to William IV, the question arose whether he could properly appear in the scarlet robes of an Oxford doctor of laws, as the color was forbidden to him.

Dalton solved this neatly: He pointed out that he was color-blind. “You call it scarlet,” he said. “To me its color is that of nature — the color of green leaves.”

# Free Falling

Published in 1869, Edward Everett Hale’s story “The Brick Moon” described the launch of an artificial satellite nearly a century before Sputnik:

If from the surface of the earth, by a gigantic peashooter, you could shoot a pea upward from Greenwich, aimed northward as well as upward; if you drove it so fast and far that when its power of ascent was exhausted, and it began to fall, it should clear the earth, and pass outside the North Pole; if you had given it sufficient power to get it half round the earth without touching, that pea would clear the earth forever. It would continue to rotate above the North Pole, above the Feejee Island place, above the South Pole and Greenwich, forever, with the impulse with which it had first cleared our atmosphere and attraction. If only we could see that pea as it revolved in that convenient orbit, then we could measure the longitude from that, as soon as we knew how high the orbit was, as well as if it were the ring of Saturn.

Because the 200-foot brick sphere is accidentally launched with human occupants, Hale perhaps also deserves credit for anticipating the space station.

# Paper Work

Rutgers mathematician E.P. Starke posed this question in the American Mathematical Monthly of July 1940:

“In high school geometry texts and elsewhere one frequently meets the statement that the reason for the straightness of the crease in a folded piece of paper is that the intersection of two planes is a straight line. This is fallacious. What is the correct reason?”

I was going to post this as a puzzle, but after much pondering I’ve been unable to make sense of the answer. Here it is:

“Let P, P′ be two points of the paper that are brought into coincidence by the process of folding. Then any point A of the crease is equidistant from P, P′, since the lines AP, AP′ are pressed into coincidence. Hence the crease, being the locus of such points A, is the perpendicular bisector of PP′.”

I agree that this is true, but I don’t see what’s wrong with the first answer. Any ideas?

UPDATE: The consensus seems to be that the first answer makes some invalid assumptions, including flat planes and Euclidean space, where Starke’s proof is more rigorous. Thanks to everyone who’s written in.

(Second update, on reflection: Presumably the books that Starke mentions were not claiming that all creases must be straight, only that a straight crease is so because two planes intersect in a line. That still seems reasonable to me.)

# Charmed

A visitor to Niels Bohr’s cottage noticed a horseshoe nailed over the door.

“Surely you don’t expect that a horseshoe will bring good luck?” asked the visitor.

“No, I don’t,” Bohr said. “But they say it works even if you don’t believe in it.”

# The Look and Say Sequence

What’s the key to this curious sequence of numbers?

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …

When read aloud, each term describes the one that precedes it. The first term consists of “one 1,” the second of “two 1s,” the third of “one 2, then one 1,” and so on.

That seems pretty arbitrary, but it opens a door into an unsuspected mathematical universe. Start with any number (except 22, an obvious dead end) and it will produce a string of digits that lengthens by about 30 percent with each generation — indeed, the percentage approaches a predictable constant (30.3577269 …) as the length approaches infinity.

More amazingly, the growing string will organize itself into a series of recognizable finite substrings that evolve predictably with each generation. John Horton Conway, who discovered all this, identified 92 such substrings, which he named after the chemical elements. Thus “uranium” (3) decays into “protactinium” (13), which becomes “thorium” (1113), and so on.

Thus an infinitely complex universe can arise from simply reading the number 1 aloud.

# The Euler Characteristic

For any convex polyhedron,

vertices – edges + faces = 2

In 1988, readers of the Mathematical Intelligencer judged this the world’s second most beautiful theorem — behind Euler’s identity.

# Misc

• Georgia, Massachusetts, and Connecticut didn’t ratify the Bill of Rights until 1939.
• Wilt Chamberlain never fouled out of a game.
• 3864 = 3 × (-8 + 64)
• What’s the opposite of “not in”?
• Alaska has a longer coastline than all other U.S. states combined.
• “To do nothing is also a good remedy.” — Hippocrates

# A Better Nature

Trisect the angles of any triangle and you’ll find an equilateral triangle at its heart.

This theorem has a curious cousin: If you trisect the sides of any triangle and erect an equilateral triangle outwardly on the middle third of each leg, then the outermost vertices of these equilateral triangles will themselves form an equilateral triangle.