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Charles Ives set himself an impossible problem [in his 114 Songs of 1922]. He wanted to use pitch distance to represent the fact that God is infinitely close to man. But what is an infinitesimally close pitch distance? In the end Ives gave up and left it to the singer to decide. Maybe what Ives wanted was a smallest perceptible pitch difference. There is no standard notation for this.

— Wilfrid Hodges, “The Geometry of Music,” in John Fauvel, ed., Music and Mathematics, 2006

The More the Merrier

anning curio

P. Anning noted this curiosity in Scripta Mathematica in 1956 — if the middle digit 1 in both the numerator and denominator of 101010101/110010011 is replaced with any odd number of 1s, then the proportion remains the same. And all of these numbers are palindromes!

Waclaw Sierpinski gives a proof in 250 Problems in Elementary Number Theory (1970).

The Bigger They Are

In 1983, University of British Columbia physicist Lorne Whitehead noted “a simple and dramatic demonstration of exponential growth, as in a nuclear chain reaction.” He determined that one domino can knock down another that’s about half again as large in all dimensions; since the gravitational potential energy of an upright domino is proportional to the fourth power of its size, this means that one tiny domino can set off a graduated chain reaction with impressively thunderous results.

Whitehead’s first domino was less than 10 mm high; he nudged it with a piece of cotton. The resulting chain reaction brought down a 13th domino that was 64 times as tall; an investment of 0.024 microjoules at one end had released 51 joules of energy at the other, an amplification factor of about 2 billion.

Of course, it’s possible to construct impressive chains of graduated dominoes even if they grow less dramatically than this one. Here’s a world record set in the Netherlands in 2009:

(Lorne A. Whitehead, “Domino ‘Chain Reaction,'” American Journal of Physics 51:2 [February 1983], 183.)


Pedro Lascuráin served as president of Mexico for less than one hour. The country’s 1857 constitution dictated that the line of succession to the presidency ran through the vice president, the attorney general, the foreign minister, and the interior minister. On Feb. 19, 1913, general Victoriano Huerta overthrew President Francisco Madero as well as his vice president and attorney general. To give his coup some appearance of legitimacy, he had foreign minister Lascuráin assume the presidency, appoint him interior minister, then resign. Lascuráin’s presidency is the shortest in world history.

Charles Brandon was Duke of Suffolk for one hour in 1551. He inherited the title when his elder brother Henry died of sweating sickness, then succumbed himself to the same disease, giving him the shortest tenure of a British peer.

Louis X of France died while his wife was pregnant, so that their son, John I, was born onto the throne. That makes him the youngest king in French history and the only person to have been king of France since birth. He lived only five days, so he’s also the only person to have held the French throne throughout his life. He’s remembered as “John the Posthumous.”

A Late Night,_mosa%C3%AFque.jpg

As a joke, Sosus of Pergamon created an “unswept floor” on which the refuse of a dinner table lies scattered.

Sosus laid the original floor at the royal palace in Pergamon in the second century B.C. It quickly became a favorite — wealthy Romans copied it, and it’s the only mosaic mentioned in Pliny’s history of art.

The Poverty Map

When H.M. Hyndman claimed that 25 percent of Londoners lived in abject poverty, Charles Booth was skeptical. So he organized his own investigation. His findings, published as Life and Labour of the People in 1889, showed that fully 35 percent of residents in the East End were poor.

In the map above, the red areas are “middle class, well-to-do,” light blue areas are “poor, 18s to 21s a week for a moderate family,” dark blue areas are “very poor, casual, chronic want,” and black areas are the “lowest class … occasional labourers, street sellers, loafers, criminals and semi-criminals.” (More of Booth’s maps can be seen here.)

A second volume, covering the rest of London, was published in 1891, and a third, in 17 volumes, appeared in 1902. He pressed for many reforms, but he remained optimistic. “What is needed is more vigorous life in every direction: social, educational, industrial, political and religious,” he concluded. “If they be evidences of vigour, pleasure seeking and extravagance need not be condemned, nor even some excess be dreaded. We may confidently trust in the balance of forces; a running stream is always wholesome; a stagnant pool, the danger.”

In a Word

adj. suitable for felling (as a tree)

Lumberjack argot, from L.G. Sorden and Jacque Vallier’s Lumberjack Lingo, 1986:

two streaks of rust: a logging railroad
cougar milk: Prohibition-era woods liquor
quinine jimmy: a camp doctor
bunch it: to quit work
kegging up: getting drunk
tree squeak: an imaginary bird to which the noise made by trees rubbing together was attributed
she’s a rainbow: What a day!
house of hesitation: a jail
traveling dandruff: lice
iron burner: the camp blacksmith
sawdust city: Eau Claire, Wisconsin

“It’s five a.m. and the gabriel blows. The bark eaters fall out of their muzzle loaders and head for the chuck house to bolt down a pile of stovelids with lots of blackstrap, some fried murphys or Johnny cake and maybe some logging berries. They dunk their rolling stock into their jerk water, growl at the hash slinger, pull up their galluses and head for the tall timber.”

Silent Warning

aa ad

In the early days of motoring, bicycle patrolmen with Britain’s Automobile Association would sometimes warn AA members of nearby speed traps. In a 1910 legal case, Chief Justice Lord Alverston found this practice illegal — in effect the patrolman would be “obstructing an officer in the course of his duty.”

So AA adopted a new policy — an AA patrolman would salute any car that bore an AA badge … unless there was a speed trap nearby. The AA Handbook warned members, “It cannot be too strongly emphasised that when a patrol fails to salute, the member should stop and ask the reason why, as it is certain that the patrol has something of importance to communicate.”

The new system was used until the 1960s.

A Cheap Pain Reliever

In a 2009 experiment, Keele University researchers Richard Stephens, John Atkins, and Andrew Kingston asked two groups of subjects to hold their hands in ice water. One group was asked to swear while they did so, and the other was asked to say neutral words. The swearers were able to hold their hands in the water twice as long, and these subjects reported feeling less pain.

No one’s quite sure why this works — perhaps swearing activates the amygdala, which leads to a release of adrenaline, producing natural pain relief. Stephens said, “I would advise people, if they hurt themselves, to swear.”

Interestingly, Stephens later found that people who reported swearing every day reported a lesser pain-deadening effect than those who swore less often. Perhaps people who seldom swear place a higher emotional value on these words, which triggers a stronger chemical response.

“Swearing is a very emotive form of language and our findings suggest that over-use of swear words can water down their emotional effect,” Stephens said. “Used in moderation, swearing can be an effective and readily available short-term pain reliever if, for example, you are in a situation where there is no access to medical care or painkillers. However, if you’re used to swearing all the time, our research suggests you won’t get the same effect.”