Napoleon’s Theorem
Image: Wikimedia Commons
Construct equilateral triangles on the sides of any triangle, and their centers will form an equilateral triangle.
This discovery is traditionally credited to Napoleon, but there’s no evidence supporting that contention. Indeed, this theorem is said to be one of the most frequently rediscovered results in mathematics.
See A Better Nature.
Rational Self-Denial
You are to choose exactly one of two opaque boxes, A and B. A mean demon has put $1,000 in the box he predicted you would not take and nothing in the other. Since you know that the predictions are quite reliable, you can be sure you will pick the wrong box. … [Now suppose] we add a small bonus for taking box B. Some of us are now inclined to say that this modification renders the A option irrational. For it seems that the bonus tips the balance that previously existed between two equally good choices. If taking box A is as rational as taking box B, then the package deal of taking B plus the bonus must be more rational than taking Box A. Yet … if the bonus makes taking B the uniquely rational choice, then you would know that the money was in box A. This knowledge would force you to change your mind in favour of taking box A.
– Roy A. Sorensen, Blindspots, 1988, after Brian Skyrms
Sorensen adds: “Perhaps this reply has some persuasiveness when the bonus is small. But now suppose that the bonus is almost as great as the prize itself, say $900. Wouldn’t it be irrational to forgo a sure $900 by taking box A?”
See Newcomb’s Paradox.
Partly Cloudy
But how are we to figure the change from ‘undecided’ to ‘true’? Is it sudden or gradual? At what moment does the statement ‘it will rain tomorrow’ begin to be true? When the first drop falls to the ground? And supposing that it will not rain, when will the statement begin to be false? Just at the end of the day, 12 p.m. sharp? … We wouldn’t know how to answer these questions; this is due not to any particular ignorance or stupidity on our part but to the fact that something has gone wrong with the way the words ‘true’ and ‘false’ are applied here.
– F. Waismann, “How I See Philosophy,” in H.D. Lewis, ed., Contemporary British Philosophy, 1956
Mirror Years
If you’re over 18, you’ve lived through two years whose dates are palindromes: 1991 and 2002. That’s a rare privilege. Since 1001, the normal gap between palindromic years has been 110 years (e.g., 1661-1771). The 11-year gap 1991-2002 has been the only exception, and we’ll wait a millennium for the next such gap, 2992-3003. Until then we’re back to 110-year intervals, and most people will see only one palindrome in a lifetime.
See Two Milestones.
Conway’s Prime-Producing Machine
Here’s something amazing — a machine made of fractions:
Start with the number 2 as your seed. Multiply it by each of the fractions above, in order, until you find one that produces an integer. (It’s 15/2.) Now adopt that integer (15) as the new seed, and multiply that by each of the fractions until you produce another integer. Keep this up, making a note whenever you produce a power of 2.
The first such power (4, or 22) appears after 19 steps. Fifty steps later, 23 turns up. Then 25 appears about 200 steps further on. A pattern emerges: the exponents are 2, 3, 5 …
It turns out that “these fourteen fractions alone have it in them to produce an infinity of primes, even those that no one yet knows about,” writes Dominic Olivastro. “There is something enormously magical about it.” John Horton Conway devised the technique; it’s an instance of his Fractran computing algorithm.
The Breaks
I once had a friend who objected to assigning chores by lot on the grounds that random selection was biased in favour of lucky people. He claimed to be serious and went on to compare unlucky people with … groups he took to be victims of discrimination. Sincere or not, wherein lies the absurdity of my friend’s objection?
– Roy A. Sorensen, Blindspots, 1988
On the House
The thirsty but impecunious soul approaches the bar-tender with a request for brandy, or what not. He takes a sip, pronounces it detestable, and offers to change it for a glass of whiskey. The obliging bar-tender substitutes the whiskey. The customer drinks, smacks his lips, and prepares to depart. ‘Here,’ says the bar-tender, ‘you haven’t paid for your whiskey.’ ‘No,’ is the innocent response; ‘I gave you the brandy in exchange for it.’ ‘But you didn’t pay for the brandy.’ ‘But I didn’t drink it.’ And while the publican intellect is vainly struggling with the mathematical puzzle involved, the puzzler makes good his escape.
– William Shepard Walsh, Handy-Book of Literary Curiosities, 1892
Misc
- EPISCOPAL is an anagram of PEPSI COLA.
- Theodore Roosevelt’s wife and mother died in the same house on the same day.
- Only a perfect square has an odd number of divisors.
- “Makes no sense makes no sense” makes no sense.
- The grounds of the Oklahoma state capitol include working oil rigs.
- “Time is the only critic without ambition.” — John Steinbeck
The Loneliest Number
Image: Wikimedia Commons
A woman approached Bertrand Russell after a lecture.
“I’m so happy to find that you’re a solipsist,” she told him. “I’ve been one all my life, and I’m surprised there aren’t more of us.”
“Boat Moved by a Rope”
There is a form of boat-racing, occasionally used at regattas, which affords a somewhat curious illustration of certain mechanical principles. The only thing supplied to the crew is a coil of rope, and they have, without leaving the boat, to propel it from one point to another as rapidly as possible. The motion is given by tying one end of the rope to the after thwart, and giving the other end a series of violent jerks in a direction parallel to the keel. I am told that in still water a pace of two or three miles an hour can be thus attained.
The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but it is not great enough to affect materially motion caused by a sufficiently big impulse. Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow n times more rapidly than it returns after the jerk, then the boat is impelled forwards at least n times more than backwards: hence on the whole the motion is forwards.
– W.W. Rouse Ball, Mathematical Recreations and Essays, 1905
Math Notes
2 × 5 × 27 = 1 × 15 × 18
2 + 5 + 27 = 1 + 15 + 18
213 × 624 = 312 × 426
102 × 402 = 201 × 204
936 × 213 = 639 × 312
Viviani’s Theorem
Image: Wikimedia Commons
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.”
See The Misfortune Field.
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.
The Thatcher Effect
When we look at another person’s face, her eyes and mouth convey the most information about her mood.
Indeed, when a face is inverted we can have trouble recognizing it because we can’t read its expression.
So in 1980 University of York psychologist Peter Thompson tried inverting everything but the eyes and mouth.
Most people can recognize the face at left and assign a mood to it, but they’re often surprised to see it right side up.
“Further research into this illusion might help determine whether face recognition is a serial or parallel process,” Thompson wrote in Perception that summer. “It might even tell us something about Margaret Thatcher.”
Scholars and Sense
O Cleinias, are those who learn the wise or the ignorant?
He answered that those who learned were the wise.
Euthydemus proceeded: There are some whom you would call teachers, are there not?
The boy assented.
And they are the teachers of those who learn — the grammar-master and the lyre master used to teach you and other boys; and you were the learners?
Yes.
And when you were learners you did not as yet know the things which you were learning?
No, he said.
And were you wise then?
No, indeed, he said.
But if you were not wise you were unlearned?
Certainly.
You then, learning what you did not know, were unlearned when you were learning?
The youth nodded assent.
Then the unlearned learn, and not the wise, Cleinias, as you imagine.
– Plato, Euthydemus
