Getting Personal
Image: Flickr

Avon, Colorado, has a bridge called Bob. The four-lane, 150-foot span, built in 1992, connects Avon with the Beaver Creek ski resort across the Eagle River. The town council held a naming contest and received 85 suggestions, including Avon Crossing and Del Mayre Bridge. It was 32-year-old construction worker Louie Sullivan who said, “Oh, heck, just name it Bob,” a suggestion that set city manager Bill James “laughing so hard he had to leave the room.”

Sullivan said he was surprised at the town’s vote; previously he had considered Avon a bit stuffy. “It raises my faith in their sense of humor,” he said.

Young Riders

Sons of Jack “Catch-‘Em-Alive” Abernathy, the youngest U.S. Marshal in history, Louis and Temple Abernathy inherited their father’s self-reliance: In 1910, when they were 10 and 6 years old, they rode on horseback from their Oklahoma ranch to Manhattan to greet Theodore Roosevelt as he returned from Africa. After riding behind Roosevelt’s car in a ticker-tape parade, they drove home in a new car.

The following year, apparently bored, they accepted a $10,000 challenge to ride on horseback from New York to San Francisco in 60 days or less, never eating or sleeping indoors. They missed the deadline by two days but still established a speed record. And in 1913 they rode by motorcycle from Oklahoma to New York City.

The two went on to successful careers in law and oil. “Teach a boy self-reliance from the moment he tumbles out of the cradle, make him keep his traces taut and work well forward in his collar, and 99 times out of a hundred his independence will assert itself before he is 2 years old,” their father told a newspaper after their first trip. “That’s my rule, and if you don’t think I’ve taken the right tack talk to my boys for five minutes and they’ll convince you that they are men in principles even if they are babies in years. God bless ’em.”

The Pythagoras Paradox

Draw a right triangle whose legs a and b each measure 1. Draw d and e to complete a unit square. Clearly d + e = 2.

Now if we cut a “step” into the square as shown, then f + h = 1 and g + i = 1, so the total length of the “staircase” is still 2. Cut still finer steps and j + k + l + m + n + o + p + q is likewise 2.

And so on: The more finely we cut the steps, the more closely their shape approximates that of the original triangle’s diagonal. Yet the total length of the stairstep shape remains 2, the sum of its horizontal and vertical elements. At the limit, then, it would seem that c must measure 2 … but we know that the length of a unit square’s diagonal is the square root of 2. Where is the error?

(Thanks, Alex.)

Mixed Greens,_After_the_earthquake_-_NARA_-_522948.tif

Professor Starr Jordan, President of Leland Stanford University, told of a case where nature had juggled with real estate during the San Francisco earthquake. An earthquake crack had passed directly in front of three cottages, and moved the rose-garden from the middle cottage to the furthest one, and the raspberry patch from the near cottage exactly opposite the middle one. History does not relate how the law decided who owned the roses and the raspberries after their rearrangement.

— M.E. David, Professor David: The Life of Sir Edgeworth David, 1937

Sea Music

The lovely Irish folk tune Port na bPúcaí (“The Music of the Fairies”) had mystical beginnings — it’s said that the people of the Blasket Islands heard ethereal music and wrote an air to match it, hoping to placate unhappy spirits. Seamus Heaney’s poem “The Given Note” tells of a fiddler who took the song “out of wind off mid-Atlantic”:

Strange noises were heard
By others who followed, bits of a tune
Coming in on loud weather
Though nothing like melody.

Recent research suggests that, rather than fairies, the islanders may have been hearing the songs of whales transmitted through the canvas hulls of their fishing boats. Humpback whales pass through Irish waters each winter as they migrate south from the North Atlantic, and their songs seem to resemble the folk tune.

Ronan Browne, who plays the air above on Irish pipes, writes, “In the mid 1990s I went rooting through some cassettes of whale song and there in the middle of the Orca (Killer Whale) section I heard the opening notes of Port na bPúcaí!”

No one can say for certain whether the one inspired the other, of course, but if it didn’t it’s certainly a pleasing coincidence.

(Thanks, James.)

A Late Contribution

A ghost co-authored a mathematics paper in 1990. When Pierre Cartier edited a Festschrift in honor of Alexander Grothendieck’s 60th birthday, Robert Thomas contributed an article that was co-signed by his recently deceased friend Thomas Trobaugh. He explained:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, ‘The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.’ Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.

Thomason himself died suddenly five years later of diabetic shock, at age 43. Perhaps the two are working again together somewhere.

(Robert Thomason and Thomas Trobaugh, “Higher Algebraic K-Theory of Schemes and of Derived Categories,” in P. Cartier et al., eds., The Grothendieck Festschrift Volume III, 1990.)

The Coin Paradox
Image: Wikimedia Commons

In the top figure, one coin rolls around another coin of equal size.

In the bottom figure, the same coin rolls along a straight line.

In each case the rolling coin has made one complete rotation. But the red arc at the top is half the length of the red line at the bottom. Why?

Round and Round

Army ants are blind; they follow the pheromone tracks left by other ants. This leaves them vulnerable to forming an “ant mill,” in which a group of ants inadvertently form a continuously rotating circle, each ant following the ones ahead and leading the ones behind. Once this happens there’s no way to break the cycle; the ants will march until they die of exhaustion.

American naturalist William Beebe once came upon a mill 365 meters in circumference, a narrow lane looping senselessly through the jungle of British Guiana. “It was a strong column, six lines wide in many places, and the ants fully believed that they were on their way to a new home, for most were carrying eggs or larvae, although many had food, including the larvae of the Painted Nest Wasplets,” he wrote in his 1921 book Edge of the Jungle. “For an hour at noon during heavy rain, the column weakened and almost disappeared, but when the sun returned, the lines rejoined, and the revolution of the vicious circle continued.”

He calculated that each ant would require 2.5 hours to make one circuit. “All the afternoon the insane circle revolved; at midnight the hosts were still moving, the second morning many had weakened and dropped their burdens, and the general pace had very appreciably slackened. But still the blind grip of instinct held them. On, on, on they must go! Always before in their nomadic life there had been a goal — a sanctuary of hollow tree, snug heart of bamboos — surely this terrible grind must end somehow. In this crisis, even the Spirit of the Army was helpless. Along the normal paths of Eciton life he could inspire endless enthusiasm, illimitable energy, but here his material units were bound upon the wheel of their perfection of instinct. Through sun and cloud, day and night, hour after hour there was found no Eciton with individual initiative enough to turn aside an ant’s breadth from the circle which he had traversed perhaps fifteen times: the masters of the jungle had become their own mental prey.”

The Paradox of the Muddy Children
Image: Wikimedia Commons

Three children return home after playing outside, and their father tells them that at least one of them has a muddy face. He repeats the phrase “Step forward if you have a muddy face” until all and only the children with muddy faces have stepped forward.

If there’s only one child with a muddy face, then she’ll step forward immediately — she can see that no other children have muddy faces, so her father must be talking about her. Each of the other children will see her muddy face and stand fast, since they have no way of knowing whether their own faces are muddy.

If there are two children with muddy faces, then no one will step forward after the first request, since each might think the father is addressing the other one. But when no one steps forward after the first request, each will realize that there must be two children with muddy faces, and that she herself must be one of them. So both will step forward after the second request, and the rest will stand fast.

A pattern emerges: If there are n children with muddy faces, then n will step forward after the nth request.

But now imagine a scenario in which more than one of the children has a muddy face, but the father does not tell them that at least one of them has a muddy face. Now no one steps forward after the first request, for the same reason as before. But no one steps forward at the second request either, because the fact that no one stepped forward after the first request no longer means that there is more than one child with a muddy face.

This is perplexing. In the second scenario all the children can see that at least one of them has a muddy face, so it seems needless for the father to tell them so. But without his statement the argument never gets going; despite his repeated requests, no child will ever step forward. What’s missing?

(From Michael Clark, Paradoxes From A to Z, 2007.)

Home Cooking

The map of the continental United States contains an elf making chicken.

He’s known as Mimal, after the states that make him up: Minnesota (hat), Iowa (head), Missouri (shirt), Arkansas (pants), and Louisiana (boots).

Fittingly, the chicken is Kentucky and the tin pan is Tennessee.