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Science & Math

Unconvinced

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In February 1962 John Glenn circled Earth three times on Friendship 7.

When he landed, he received a card from the International Flat Earth Research Society.

It said, “OK wise guy.”

Insight

In 1945, Oxford University’s Museum of the History of Science realized that 14 astrolabes were missing from its collection. Curator Robert T. Gunther had arranged for storage of the museum’s objects during the war, but both he and the janitor who had helped him had died in 1940. The missing instruments, the finest of the museum’s ancient and medieval astrolabes, were irreplaceable, the only examples of their kind. Where had Gunther hidden them?

The museum consulted the Oxford city police and Scotland Yard, who searched basements and storerooms throughout the city. The Times, the Daily Mail, and the Thames Gazette publicized the story. Inquiries were extended to local taxi drivers and 108 country houses. At Folly Bridge, Gunther’s house, walls were inspected, flagstones lifted, and wainscoting prised away. A medium and a sensitive were even consulted, to no avail. Finally the detective inspector in charge of the case reviewed the evidence and composed a psychological profile of Gunther, a man he had never met:

Clever professor type, a bit irascible, who didn’t get on too well with his colleagues. Single minded. Lived for the Museum. Hobby in Who’s Who ‘… founding a Museum’. Used to gloat over the exhibits and looked upon them as his own creation. Never allowed anyone else to handle them. Reticent, even secretive. Never told anyone what he what he was going to do. Didn’t trust them, perhaps. Not even his friends the Rumens, who would have offered their car to move the things. Had original ideas though. Safe from blast below street level. Germans would never bomb Oxford. Why, its total war damage was £100 and that from one of our own shells. How right he was. He never expected to die then. Believed he’d live to 90. Hadn’t made any plans; like most of us he thought he might get bumped off when the war started. That’s what he was telling his son in those letters. There was only one conclusion with a man like that anyhow: he’d never let the things out of his reach if he could have helped it. Didn’t even take the trouble to pack his own treasures away in Folly Bridge.

In 1948 the new curator found the missing instruments — they were right “within reach” in the museum’s basement. Gunther had disguised their crate with a label reading “Eighteenth-Century Sundials,” and it had evaded detection throughout the searches.

From A.E. Gunther, Early Science in Oxford, vol. XV, 1967, 303-309.

(Thanks, Diane.)

Art Appreciation

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If we stand immediately below a painting in a gallery, it appears foreshortened. But if we stand on the other side of the room, it appears small. Somewhere between these two points must be the optimum viewing position, where the painting fills the widest possible angle in our vision. How can we find it?

The German mathematician Regiomontanus posed this question in 1471. We can solve it using calculus, but it also yields to simple geometry: Draw a circle defined by the top and bottom of the painting and our eye level. That’s the point we want — any other point at eye level will define a larger circle, in which the picture makes a smaller chord and subtends a smaller angle.

(Thanks, David.)

Equal Opportunity

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The Pythagorean theorem works for any similar shapes, not just squares.

In the figure above, A + B = C.

If the three sides of a right triangle are made the diameters of three circles, then the combined area of the two smaller circles equals that of the largest. That’s also the area of the circumcircle, since a right triangle’s hypotenuse forms the diameter of its circumscribing circle.

Digit Count

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A letter from John Phillips of the Yale University School of Medicine to the New England Journal of Medicine, Feb. 14, 1991:

When referring to the hand, the names digitus pollicis, indicis, medius, annularis, and minimus specify the five fingers. In situations of clinical relevance the use of such names can preclude anatomical ambiguity. These time-tested terms have honored the fingers, but the toes have been labeled only by number, except of course the great toe, or hallux. Is it not time for the medical community to have the toes no longer stand up and merely be counted? I submit for consideration the following nomenclature to refer to the pedal digits: for the hallux, porcellus fori; for the second toe, p. domi; for the third toe, p. carnivorus; for the fourth toe, p. non voratus; and for the fifth toe, p. plorans domum.

Using porcellus as the diminutive form of porcus, or pig, one can translate the suggested terminology as follows: piglet at market, piglet at home, meat-eating piglet, piglet having not eaten, and piglet crying homeward, respectively.

(Thanks, Scott.)

Words and Numbers

alphamagic square 1

Spell out each number in this magic square and count its letters (25 -> TWENTY-FIVE -> 10), and you’ll produce another magic square:

alphamagic square 2

Lee Sallows extends this idea into geometry:

Casket Trouble

“If Socrates died, he died either when he was alive or when he was dead. He did not die when he was alive — for then the same man would have been both living and dead. Nor when he was dead; for then he would have been dead twice. Therefore Socrates did not die.”

– Sextus Empiricus, Against the Physicists

Going Down

In antiquity Aristotle had taught that a heavy weight falls faster than a light one. In 1638, without any experimentation, Galileo saw that this could not be true. What had he realized?

Click for Answer

Double Indemnity

This past February, brothers named Elwin and Yohan were arrested for six rapes in France, but both denied the charges. Deciding which is guilty is a tricky affair — they’re identical twins, so the genetic difference between them is very slight. Marseille police chief Emmanual Kiehl said, “It could take thousands of separate tests before we know which one of them may be guilty.”

This is only the latest in a series of legal conundrums involving identical twins and DNA evidence. During a jewel heist in Germany in January 2009, thieves left behind a drop of sweat on a latex glove. A crime database showed two hits — identical twins Hassan and Abbas O. (under German law their last name was withheld). Both brothers had criminal records for theft and fraud, but both were released. The court ruled, “From the evidence we have, we can deduce that at least one of the brothers took part in the crime, but it has not been possible to determine which one.”

Later that year, identical twins Sathis Raj and Sabarish Raj escaped hanging in Malaysia when a judge ruled it was impossible to determine which was guilty of drug smuggling. “Although one of them must be called to enter a defence, I can’t be calling the wrong twin to enter his defence,” the judge told the court. “I also can’t be sending the wrong person to the gallows.”

In 2003, a Missouri woman had sex with identical twins Raymon and Richard Miller within hours of one another. When she became pregnant, both men denied fathering the child. In Missouri a man can be named a legal father only if a paternity test shows a 98 percent or higher probability of a DNA match, but the Miller twins both showed a probability of more than 99.9 percent.

“With identical twins, even if you sequenced their whole genome you wouldn’t find difference,” forensic scientist Bob Gaensslen told ABC News at the time. More recent research shows that this isn’t the case, but teasing out the difference can be expensive — in the Marseilles case, police were told that such a test would cost £850,000.

It goes on. Last month British authorities were trying to decide how to prosecute a rape when DNA evidence identified both Mohammed and Aftab Asghar. “It is an unusual case,” said prosecutor Sandra Beck. “They are identical twins. The allegation is one of rape. There is further work due.”

Misc

  • Fathers can mother, but mothers can’t father.
  • The Mall of America is owned by Canadians.
  • Neil Armstrong was 17 when Orville Wright died.
  • LONELY TYLENOL is a palindrome.
  • 258402 + 437762 = 2584043776
  • “The mind is not a vessel to be filled, but a fire to be kindled.” — Plutarch

Edward Gorey’s pen names included Ogdred Weary, Raddory Gewe, Regera Dowdy, D. Awd­rey-Gore, E.G. Deadworry, Waredo Dyrge, Deary Rewdgo, Dewda Yorger, and Dogear Wryde. Writer Wim Tigges responded, “God reward ye!”

Thought Chemistry

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How many ideas hover dispersed in my head of which many a pair, if they should come together, could bring about the greatest of discoveries! But they lie as far apart as Goslar sulphur from East India saltpeter, and both from the dust in the charcoal piles on the Eichsfeld — which three together would make gunpowder. How long the ingredients of gunpowder existed before gunpowder did! There is no natural aqua regia. If, when thinking, we yield too freely to the natural combinations of the forms of understanding and of reason, then our concepts often stick so much to others that they can’t unite with those to which they really belong. If only there were something in that realm like a solution in chemistry, where the individual parts float about, lightly suspended, and thus can follow any current. But since this isn’t possible, we must deliberately bring things into contact with each other. We must experiment with ideas.

– G.C. Lichtenberg, Aphorisms

Almost Home

A drunk man arrives at his doorstep and tries to unlock his door. There are 10 keys on his key ring, one of which will fit the lock. Being drunk, he doesn’t approach the problem systematically; if a given key fails to work, he returns it to the ring and then draws again from all 10 possibilities. He tries this over and over until he gets the door open. Which try is most likely to open the door?

Surprisingly, the first try is most likely. The probability of choosing the right key on the first try is 1/10. Succeeding in exactly two trials requires being wrong on the first trial and right on the second, which is less likely: 9/10 × 1/10. And succeeding in exactly three trials is even less likely, for the same reason. The probability diminishes with each trial.

“In other words, it is most likely that he will get the right key at the very first attempt, even if he is drunk,” writes Mark Chang in Paradoxology of Scientific Inference. “What a surprise!”

A Losing Game

You and I each have a stack of coins. We agree to compare the coins atop our stacks and assign a reward according to the following rules:

  • If head-head appears, I win $9 from you.
  • If tail-tail appears, I win $1 from you.
  • If head-tail or tail-head appears, you win $5 from me.

After the first round each of us discards his top coin, revealing the next coin in the stack, and we evaluate this new outcome according to the same rules. And so on, working our way down through the stacks.

This seems fair. There are four possible outcomes, all equally likely, and the payouts appear to be weighted so that in the long run we’ll both break even. But in fact you can arrange your stack so as to win 80 cents per round on average, no matter what I do.

Let t represent the fraction of your coins that display heads. If my coins are all heads, then your gain is given by

GH = -9t + 5(1 – t) = -14t + 5.

If my coins are all tails, then your gain is

GT = +5t – 1(1 – t) = 6t – 1.

If we let GH = GT, we get t = 0.3, and you gain GH = GT = $0.80.

This result applies to an entire stack or to any intermediate segment, which means that it works even if my stack is a mix of heads and tails. If you arrange your stack so that 3/10 of the coins, randomly distributed in the stack, display heads, then in a long sequence of rounds you’ll win 80 cents per round, no matter how I arrange my own stack.

(From J.P. Marques de Sá, Chance: The Life of Games & the Game of Life, 2008.)

“On Professor Freud”

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The ignorant pronounce it Frood,
To cavil or applaud.
The well-informed pronounce it Froyd,
But I pronounce it Fraud.

– G.K. Chesterton

Higher Things

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In 1966 a Swedish encyclopedia publisher requested a photograph of Richard Feynman “beating a drum” to give “a human approach to a presentation of the difficult matter that theoretical physics represents.” Feynman responded:

Dear Sir,

The fact that I beat a drum has nothing to do with the fact that I do theoretical physics. Theoretical physics is a human endeavor, one of the higher developments of human beings, and the perpetual desire to prove that people who do it are human by showing that they do other things that a few other human beings do (like playing bongo drums) is insulting to me.

I am human enough to tell you to go to hell.

Yours,

RPF

Single Cases

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If we roll a fair die an infinite number of times, the outcome 4 occurs in 1/6 of the cases. In this light we can say that the probability of rolling a 4 with this die is 1/6. But suppose that, instead of repeating the experiment forever, we roll the die only once. Now it still seems natural to say that there’s a 1/6 chance of rolling a 4, but in fact either we’ll roll a 4 … or we won’t. Can it make sense to assign a probability to a single outcome? Charles Sanders Peirce writes:

If a man had to choose between drawing a card from a pack containing twenty-five red cards and a black one, or from a pack containing twenty-five black cards and a red one, and if the drawing of a red card were destined to transport him to eternal felicity, and that of a black one to consign him to everlasting woe, it would be folly to deny that he ought to prefer the pack containing the larger proportion of red cards, although, from the nature of the risk, it could not be repeated. It is not easy to reconcile this with our analysis of the conception of chance. But suppose he should choose the red pack, and should draw the wrong card, what consolation would he have? He might say that he had acted in accordance with reason, but that would only show that his reason was absolutely worthless. And if he should choose the right card, how could he regard it as anything but a happy accident? He could not say that if he had drawn from the other pack, he might have drawn the wrong one, because an hypothetical proposition such as, ‘if A, then B,’ means nothing with reference to a single case.

Peirce’s solution to this problem is curiously humanistic. Our inferences must extend to include the interests of all races in all epochs. A soldier storms a fort knowing that he may die but that his zeal, if carried through the regiment, will win the day. The man trying to draw a red card “cannot be logical so long as he is concerned only with his own fate” but “should care equally for what was to happen in all possible cases … and would draw from the pack with the most red cards.”

“He who would not sacrifice his own soul to save the whole world, is, as it seems to me, illogical in all his inferences, collectively.”

Pursuit of Truth

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Can animals reason without using language? Sextus Empiricus writes:

[Chrysippus] declares that the dog makes use of the fifth complex indemonstrable syllogism when, on arriving at a spot where three ways meet …, after smelling at the two roads by which the quarry did not pass, he rushes off at once by the third without stopping to smell. For, says the old writer, the dog implicitly reasons thus: ‘The animal went either by this road, or by that, or by the other: but it did not go by this or that, therefore he went the other way.’

So, perhaps. There’s a limit, though.

The Lakes of Wada

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Find a square island and establish a blue lake on it, bringing blue water within a certain distance of every point on the island’s remaining dry land. Then create a red lake, bringing red water even closer to every point on the remaining land, and a green lake bringing green water still closer.

If you continue this indefinitely, irrigating the island more and more aggressively from each lake in turn, you’ll reach the perplexing state where the three lakes have the same boundary. Japanese mathematician Kunizo Yoneyama offered this example in 1917.

Miquel’s Theorem

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Draw any triangle, pick a point on each side, and connect these in pairs to the vertices using circles as shown.

The circles will always intersect in a single point.

Further, the angles marked in green will all be equal.

Jury Duty

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From Gábor J. Székely’s Paradoxes in Probability Theory and Mathematical Statistics, via Mark Chang’s Paradoxology of Scientific Inference:

A, B, C, D, and E make up a five-member jury. They’ll decide the guilt of a prisoner by a simple majority vote. The probability that A gives the wrong verdict is 5%; for B, C, and D it’s 10%; for E it’s 20%. When the five jurors vote independently, the probability that they’ll bring in the wrong verdict is about 1%. But if E (whose judgment is poorest) abandons his autonomy and echoes the vote of A (whose judgment is best), the chance of an error rises to 1.5%.

Even more surprisingly, if B, C, D, and E all follow A, then the chance of a bad verdict rises to 5%, five times worse than if they vote independently, even though A is nominally the best leader. Chang writes, “This paradox implies it is better to have your own opinion even if it is not as good as the leader’s opinion, in general.”

Possession

When in very good spirits he would jest in a delightful manner. This took the form of deliberately absurd or extravagant remarks uttered in a tone, and with a mien, of affected seriousness. On one walk he ‘gave’ to me each tree that we passed, with the reservation that I was not to cut it down or do anything to it, or prevent the previous owners from doing anything to it: with those reservations it was henceforth mine. Once when we were walking across Jesus Green at night, he pointed at Cassiopeia and said that it was a ‘W’ and that it meant Wittgenstein. I said that I thought it was an ‘M’ written upside down and that it meant Malcolm. He gravely assured me that I was wrong.

– Norman Malcolm, Ludwig Wittgenstein: A Memoir, 1958

Special Thanks

Botanist George B. Hinton named the plant species Salvia leninae Epling after a saddle mule, Lenina, who had helped him to gather more than 150,000 specimens in the mountains of western Mexico.

He wrote, “What is more deserving of commemoration than the dignity of long and faithful service to science, even though it be somewhat unwitting — or even unwilling?”

See Rigged Latin.

Quick Thinking

In summer 1940, Germany demanded access to Swedish telephone cables to send encoded messages from occupied Norway back to the homeland. Sweden acceded but tapped the lines and discovered that a new cryptographic system was being used. The Geheimschreiber, with more than 800 quadrillion settings, was conveying top-secret information but seemed immune to a successful codebreaking attack.

The Swedish intelligence service assigned mathematician Arne Beurling to the task, giving him only a pile of coded messages and no knowledge of the mechanism that had been used to encode them. But after two weeks alone with a pencil and paper he announced that the G-schreiber contained 10 wheels, with a different number of positions on each wheel, and described how a complementary machine could be built to decode the messages.

Thanks to his work, Swedish officials learned in advance of the impending invasion of the Soviet Union. Unfortunately, Stalin’s staff disregarded their warnings.

“To this day no one knows exactly how Beurling reasoned during the two weeks he spent on the G-Schreiber,” writes Peter Jones in his foreword to The Codebreakers, Bengt Beckman’s account of the exploit. “In 1976 he was interviewed about his work by a group from the Swedish military, and became extremely irritated when pressed for an explanation. He finally responded, ‘A magician does not reveal his tricks.’ It seems the only clue Beurling ever offered was the remark, cryptic itself, that threes and fives were important.”

(Thanks, John.)

Fore!

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In 1962 mycologist R.W.G. Dennis reported a new species of fungus he had observed growing in Lancashire and East Africa. He called it Golfballia ambusta:

The unopened fruit body evidently closely resembles certain small, hard but elastic, spheres employed by the Caledonians in certain tribal rites, practised at all seasons of the year in enclosures of partially mown grass set apart for the purpose. The diameter of the volva is approximately 3 cm., its surface smooth or regularly furrowed, becoming much wrinkled after dehiscence, its texture extremely hard and tough. A gelatinous stratum, so characteristic of other phalloids, is wanting. The appearance and texture of the immature gleba is still unknown but at maturity it is extruded as a column, thickly set with short strap-like processes of an elastic consistency, each scarcely 1 cm. long and 1.5 mm. wide, abruptly truncated at the free end. As with other phalloids, there is a strong and distinctive odour, in this instance not unpleasant and identified independently by several observers as reminiscent of old or heated india-rubber. This is probably a reliable and important diagnostic character. Taste not recorded but probably mild; the fruit bodies are unlikely to be toxic but may well prove inedible from their texture. Spores have not been recovered and the means of reproduction therefore remains unknown.

It seems to be very prolific in America as well.

(R.W.G. Dennis. A remarkable new genus of phalloid in Lancashire and East Africa, Journ. Kew Guild. 8, 67 (1962): 181-182.)