The Perfect Box
Image: Wikimedia Commons

Certainly rectangular cuboids exist whose edges and face diagonals all have integer lengths.

For example, in 1719 Paul Halcke discovered one with edges (a, b, c) = (44, 117, 240) and face diagonals (d, e, f ) = (125, 244, 267).

But does one exist whose space diagonal (here shown in red) also has integer length?

As of last September, none has been found and no one has proven that none exist.

A Portable Bed

In 1919, Mrs. Ray Werner patented a military overcoat with an inflatable lining:

The primary object of my invention is to provide a garment, the back of which may be inflated to provide a resilient support for the body of the wearer without removing the garment, thus providing a greater comfort while reclining in a recumbent position.

A separate compartment can be inflated into a pillow. The application was granted that September; I don’t know whether it was ever manufactured.


Devised in 1948 by Russian choreographer Nadezhda Nadezhdina, the “floating step” of the Beryozka Dance Ensemble seems to carry dancers smoothly across the stage, as though their feet have left the ground.

“Not even all our dancers can do it,” Nadezhdina said. “You have to move in very small steps on very low half-toe with the body held in a certain corresponding position.”


I send you a photo of myself for ‘Curiosities’ studying Virgil in a peculiar position. It was taken by my brother in the country a few weeks ago. It was not a snap-shot, but a time exposure.

— Charles E. Williams of Rock Ferry, Cheshire, in Strand, September 1905

Real Time

Christian Marclay’s 2010 film The Clock is a montage of 12,000 film clips that depict clocks or timepieces, arranged in chronological order.

It’s designed to run in a continuous 24-hour loop — so the film itself functions as a clock.

For the Record

When a tornado struck Mayfield, Ohio, in 1842, Western Reserve College mathematician Elias Loomis noticed that several fowl had been picked almost clean of their feathers. To find out what wind velocity could accomplish this, he charged a cannon with 5 ounces of gunpowder and inserted a freshly killed chicken in place of a ball:

As the gun was small, it was necessary to press down the chicken with considerable force, by which means it was probably somewhat bruised. The gun was pointed vertically upwards and fired; the feathers rose twenty or thirty feet, and were scattered by the wind. On examination they were found to be pulled out clean, the skin seldom adhering to them. The body was torn into small fragments, only a part of which could be found. The velocity is computed at five hundred feet per second, or three hundred and forty one miles per hour. A fowl, then, forced through the air with this velocity, is torn entirely to pieces; with a less velocity, it is probable most of the feathers might be pulled out without mutilating the body.

“If I could have the use of a suitable gun I would determine this velocity by experiment,” he ended. “It is presumed to be not far from a hundred miles per hour.”

(Elias Loomis, “On a Tornado Which Passed Over Mayfield, Ohio, February 4th, 1842,” American Journal of Science 43:2 [July-September 1842], 278-301.)

The Paradox of the Just Law

How can a just law have a claim on our obedience? Murder is wrong, regardless of what the law says about it; we expect people to refrain from murder because it’s wrong, not because it’s prohibited or punished. I’d be offended if someone suggested that it’s only respect for the law that’s restraining me from committing murder. But this suggests that we don’t have an obligation to obey laws that prohibit murder — a morally conscientious person should never find himself obliged to submit to them.

“The more just and valuable the law is … the more reason one has to conform to it, and the less to obey it,” writes legal philosopher Joseph Raz. “Since it is just, those considerations which establish its justice should be one’s reasons for conforming with it, i.e., for acting as it requires. But in acting for these reasons one would not be obeying the law, one would not be conforming because that is what the law requires.”

(Note, though, that Raz says the paradox is only apparent — see his full paper here.)

(Scott Hershovitz, “The Authority of Law,” in Andrei Marmor, ed., The Routledge Companion to Philosophy of Law, 2012.)

Shared Birthdays

Famously, in a group of 23 randomly chosen people, the chance is slightly higher than 50 percent that two will share a birthday.

In 2014, James Fletcher considered the birth dates of players in the World Cup, who were conveniently organized into squads of 23 people each. He found that 16 of the 32 squads had at least one shared birthday. If data from 2010 World Cup was included, 31 of 64 squads had shared birthdays, still quite close to 50 percent.

If a group numbers 366 people, the probability of a shared birthday is 100 percent (neglecting leap years). But to reach 99 percent certainty we need only 55 people. “It is almost unbelievable that such a small difference between the probabilities 99% and 100% can lead to such a big difference between the numbers of people,” writes Gabor Szekely in Paradoxes in Probability Theory and Mathematical Statistics (1986). “This paradoxical phenomenon is one of the main reasons why probability theory is so wide-ranging in its application.”

Justified Killing

Self-defense is widely accepted as a valid reason to use deadly force. But why is it valid? Most other kinds of killing arouse strong moral and political controversy: capital punishment, abortion, euthanasia, suicide, war, even the killing of animals. But debates about self-defense tend to accept its basic legitimacy. Even those who oppose national self-defense as a justification for war may accept the same principle in a conflict between individuals.

“Not only is self-defense uniquely uncontroversial as a form of killing, but the lack of controversy persists despite the absence of any plausible account as to why it is justified,” writes philosopher Whitley R.P. Kaufman in Justified Killing (2009). “The strength and unanimity with which the assumption that killing in self-defense is morally and legally permissible is held suggest that there must be some powerful and persuasive rationale justifying such killing. But if there is such a rationale, moral philosophy has yet to find it.”

Slow Burn

Australia’s Mount Wingen has been burning for 6,000 years. A lightning strike or brush fire ignited a coal seam there around 4000 B.C., and it’s been smoldering ever since.

“Smoldering fires, the slow, low-temperature, flameless form of combustion, are an important phenomena in the Earth system, and the most persistent type of combustion,” University of Edinburgh fire scientist Guillermo Rein told the New York Times. “The most important fuels involved in smoldering fires are coal and peat. Once ignited, these fires are particularly difficult to extinguish despite extensive rains, weather changes or firefighting attempts, and can persist for long periods of time (months, years), spreading deep (5 meters) and over extensive areas of forest subsurface. Indeed, smoldering fires are the longest continuously burning fires on Earth.”

The Mount Wingen fire holds the Guinness record for the longest-burning fire in the world.