More chess masters reside in New York City than in the rest of the United States combined. We’re planning a chess tournament that all American masters are expected to attend, and we want to minimize the total intercity traveling done by the players. The New York players argue that, by this criterion, the tournament should be held in their city. The West Coast players argue that a city should be chosen near the center of the gravity of the players. Where should we hold the tournament?
There are no known pictures of two American presidents’ wives: Martha Jefferson and Margaret Taylor.
We have one silhouette (left) of Jefferson, who was a little over 5 feet tall and had auburn hair and hazel eyes.
And one 1903 book contains a suggested likeness of Taylor (right), who was described during her life as “a fat, motherly looking woman,” “countenance rather stern but it may be the consequence of military association.”
But no portrait of either woman is known to exist. Some artists have attempted renderings based on pictures of their daughters, whom they were said to resemble, but that’s the best we can do.
In his 1984 book Scaling, Duke University physiologist Knut Schmidt-Nielsen points out a pleasing coincidence:
A 30-gram mouse that breathes at a rate of 150 times per minute will breathe about 200 million times during its 3-year life; a 5-ton elephant that breathes at the rate of 6 times per minute will take approximately the same number of breaths during its 40-year lifespan. The heart of the mouse, ticking away at 600 beats per minute, will give the mouse some 800 million heartbeats in its lifetime. The elephant, with its heart beating 30 times per minute, is awarded the same number of heartbeats during its life.
In fact, most mammals have roughly the same number of heartbeats per lifetime, about 109. Small mammals have high metabolic rates and short lives; large ones have low rates and long lives. Humans are lucky: “We live several times as long as our body size suggests we should.”
excogitous
adj. inventive
volitorial
adj. pertaining to flying
empyreuma
n. a burnt smell
Newsreel men recently witnessed an unscheduled drama as flames ended the attempt of Constantinos Vlachos, co-inventor of one of the strangest of flying craft, to win government aid for its development. He had planned an ascent from the lawn of the Congressional Library at Washington, D.C., to demonstrate his ‘triphibian,’ which he claimed could navigate in the air, on land, or in the water. Hardly had he started the motor when fire enveloped the machine. Spectators dashed to his aid and dragged him, severely burned, from the blazing wreck.
— Popular Science, January 1936
(Thanks, Tucker.)
Is it possible to sail on a river on a windless day? In Why Cats Land on Their Feet (2012), Mark Levi points out that the answer is yes, at least in principle. If the keel is turned broadly against the current, then this will carry the boat downstream, drawing the sail through the still air. Now the roles of the sail and the keel are reversed: The keel catches the motion of the river, acting as a sail, and the boat follows the course established by the sail, which acts as a keel. “It’s just like regular sailing,” Levi writes, “except upside down.”
That’s from the point of view of an observer on shore. In the boat’s reference frame, the water is still and a wind is blowing upstream. From this perspective the boat is sailing conventionally — the sail is catching the wind and the keel slices through the water.
“This is a neat symmetry,” Levi notes. “The sail and the keel exchange roles, depending on your reference frame! So the sail and the keel have completely equal rights in that respect.”
“How much more grievous are the consequences of anger than the causes of it.” — Marcus Aurelius
“When anger rises, think of the consequences.” — Confucius
“Whatever is begun in anger ends in shame.” — Benjamin Franklin
The diagram above represents a road network where T is the number of travelers. The leg from START to A takes T/100 = t minutes to traverse, as does the leg from B to END. The legs from START to B and from A to END each take a constant 45 minutes.
Now suppose that 4000 drivers want to travel from START to END. The northern and the southern routes are equally efficient, so the drivers will split into two groups, and each will arrive at END in 2000/100 + 45 = 65 minutes.
But now suppose that planners, hoping to improve matters, add a shortcut between A and B with a travel time of 0 minutes. Now all the drivers will take the route from START to A, since in the worst case it will take 4000/100 = 40 minutes, rather than the guaranteed 45 minutes taken by the leg from START to B. From A every driver will take the shortcut to B, for the same reason: Even in the worst case, the trip from B to END is 5 minutes faster than the trip from A to END.
As a result, every driver’s trip now takes 4000/100 + 4000/100 = 80 minutes, which is 15 minutes longer than in the original state of affairs. No individual driver has an incentive to change his behavior, since now the two original routes (northern and southern) each take 4000/100 + 45 = 85 minutes. If the 4000 drivers as a body could agree never to use the shortcut, they’d all be better off. But without a way to enforce this, all are stuck with longer commutes.
The principle was discovered by German mathematician Dietrich Braess in 1968. It’s known as Braess’ paradox.
GIROLAMO SAVONAROLA alternates vowels and consonants.
(Thanks, Joseph.)
On June 23, 1858, the Catholic Church removed 6-year-old Edgardo Mortara from his family in Bologna. The reason they gave was surprising: The Mortaras were Jewish, and Edgardo had been secretly baptized. In this week’s episode of the Futility Closet podcast we’ll tell the story of little Edgardo and learn how his family’s plight shaped the course of Italian history.
We’ll also hear Ben Franklin’s musings on cultural bigotry and puzzle over an unexpected soccer riot.
