Art Appreciation

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

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

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.)

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.”


  • 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

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!”