Here’s a subtly impossible figure devised by UC-Santa Cruz computer scientist David Huffman. If it’s a three-sided pyramid, then its edges define the intersections of three planes and should meet in a single point. But they don’t:
This is intriguing because the figure doesn’t immediately look impossible. In Vagueness and Contradiction, philosopher Roy Sorensen writes, “The impossibility of an appearance is sometimes concealed without overloading our critical capacities.”
Possibly this is because we sense that other solutions are possible that can reconcile the error. Zenon Kulpa points out that the pyramid becomes intelligible if we imagine that the farther side hides a fourth edge, giving the figure four sides rather than three. He describes two families of such solutions in “Are Impossible Figures Possible?”, Signal Processing, May 1983.
In 1935 a shark in an Australian aquarium vomited up a human forearm, a bizarre turn of events that sparked a confused murder investigation. This week’s episode of the Futility Closet podcast presents two cases in which a shark supplied key evidence of a human crime.
We’ll also learn about the Paris Herald’s obsession with centigrade temperature, revisit the scary travel writings of Victorian children’s author Favell Lee Mortimer, and puzzle over an unavenged killing at a sporting event.
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.
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.”
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?
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
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.
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.
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.)
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.”
