Mathematician Paul Erdős had no home, no job, and no hobbies. Instead, for 60 years he wandered the world, staying with each of hundreds of collaborators just long enough to finish a project, and then moving on. In this week’s episode of the Futility Closet podcast we’ll meet the “magician of Budapest,” whose restless brilliance made him the most prolific mathematician of the 20th century.
We’ll also ponder Japanese cannibalism in World War II and puzzle over a senseless stabbing.
A low-tech model of human cognition, from William James’ The Principles of Psychology, 1890:
If we make a solid wooden frame with the sentence written on its front, and the time-scale on one of its sides, if we spread flatly a sheet of India rubber over its top, on which rectangular co-ordinates are painted, and slide a smooth ball under the rubber in the direction from 0 to ‘yesterday,’ the bulging of the membrane along this diagonal at successive moments will symbolize the changing of the thought’s content in a way plain enough, after what has been said, to call for no more explanation. Or to express it in cerebral terms, it will show the relative intensities, at successive moments, of the several nerve-processes to which the various parts of the thought-object correspond.
He was grappling with the stream of consciousness, the notion that thought is a flowing stream rather than a distinct chain of ideas, and with the realization that studying this by introspection is ultimately futile: “The rush of thought is so headlong that it almost always brings us up at the conclusion before we can arrest it. Or if our purpose is nimble enough and we do arrest it, it ceases forthwith to be itself. … The attempt at introspective analysis in these cases is in fact like seizing a spinning top to catch the motion, or trying to turn up the gas quickly enough to see how the darkness looks.”
In 1993, cognitive neuroscientist Stanislas Dehaene asked respondents to classify a number as larger or smaller than 65, using response keys held in their hands. Interestingly, the subjects who held the “smaller” key in their left hand and the “larger” key in their right responded more quickly and with fewer errors than those in the opposite group. This suggests that we carry around a mental number line in our heads, implicitly associating left with “small” and right with “large”; the subjects in the slower group may have been fighting against this prejudice. Dehaene calls this the SNARC effect, for “spatial-numerical association of response codes.”
The effect was borne out in later studies. When subjects were asked to cross their arms, the group whose “smaller” button lay to their left were still faster than their counterparts. And the effect still obtains regardless of the range of numbers used, and even in tasks where the size of the number is irrelevant: In another experiment subjects were asked to report whether a given number was odd or even; here again, responses to numbers in the upper half of the test range were quicker when the appropriate response key was on the right, and likewise for small numbers on the left.
Interestingly, Iranian students living in France who had initially learned to read from right to left showed a reverse SNARC effect (associating small numbers with the right and large numbers with the left) if they’d recently immigrated, but those who had lived in France for some time showed the same SNARC effect as native French students.
“Very probably, then, this number-space association is learned, not innate,” writes M. Giaquinto in Visual Thinking in Mathematics. “But there may very well be an innate propensity in operation here. A left-right association has been found for familiar ordered sets of non-numerical items, namely, months and letters. This suggests that we have a tendency to form a linear spatial representation of any set of things whose customary presentation is well ordered (in the mathematical sense).”
(S. Dehaene, S. Bossini, and P. Giraux, “The Mental Representation of Parity and Numerical Magnitude,” Journal of Experimental Psychology: General 122, 371-396. See Number Forms.)
Somewhat like Eratosthenes, the Iranian polymath Al-Biruni (973-1048) was able to estimate the radius of the earth using just a few measurements and some clear thinking. If h is a mountain with a known height and the distance from the mountaintop A to the horizon C can be established accurately, then angle α is the same as angle AOC at the earth’s center and we have everything we need to calculate leg OC of right triangle AOC, which is the radius of the earth.
Biruni carried this out using a tall mountain near Nandana in present-day Pakistan. He estimated the earth’s radius at 6,339.9 km, which is only 16.8 km less than the current value of 6,356.7 km. This accuracy would not be obtained in the West until the 16th century.
06/22/2017 UPDATE: Wait, he didn’t even need the distance to the horizon, just the mountain height and the dip angle. Details here. (Thanks, Jacob.)
In a diary entry in 1843, Sir Oswald Brierly, manager of the whaling station at Twofold Bay in southeast Australia, noted a strange cooperative relationship that had grown up between killer whales and the local whalers:
They [the killer whales] attack the [humpback] whales in packs and seem to enter keenly into the sport, plunging about the [whaling] boat and generally preventing the whale from escaping by confusing and meeting him at every turn. … The whalemen of Twofold Bay are very favourably disposed towards the killers and regard it as a good sign when they see a whale ‘hove to’ by these animals because they regard it as an easy prey when assisted by their allies the killers.
By the early 20th century this curious custom had grown into a complex operation. The killer whales would herd a passing humpback into the bay and harass it there while others swam to the whaling station, breached, and thrashed their tails to alert the whalers. When the whalers arrived and harpooned the humpback, the killers would continue to leap onto its back and blowhole to tire it. In return, the whalers would anchor the dead whale to the bottom for a day or two so that the killers could feast on its lips and tongue.
The whalers came to know many of these killer whales by name: Hooky, Cooper, Typee, Jackson, and so on. The most famous, Old Tom, worked with the Twofold Bay whalers for almost four decades in the early 20th century — he grew famous for gripping the harpoon line with his teeth as each doomed humpback towed the whaleboat through the water. He died in 1930, and his skeleton, complete with grooves in the teeth, now resides in the Eden Killer Whale Museum in New South Wales.
(From Hal Whitehead and Luke Rendell, The Cultural Lives of Whales and Dolphins, 2014. See A Feathered Maître d’.)
In the Fibonacci sequence, each number is the sum of the two preceding ones:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, …
This produces two notable secondary patterns: Summing the squares of each pair of adjacent entries yields an even-numbered term in the sequence:
12 + 12 = 2
12 + 22 = 5
22 + 32 = 13
32 + 52 = 34
52 + 82 = 89
82 + 132 = 233
132 + 212 = 610
And the odd-numbered terms between these are the differences of squares of terms taken two by two, two places apart:
22 – 12 = 3
32 – 12 = 8
52 – 22 = 21
82 – 32 = 55
132 – 52 = 144
212 – 82 = 377
342 – 132 = 987
… and so on.
In 1988, workers found a 5-year-old bottlenose dolphin trapped in a canal lock in South Australia. She was taken to a commercial aquarium for medical treatment, where she was named Billie and housed with the trained dolphins who performed for oceanarium visitors there. One of the behaviors that the trained dolphins had learned was tailwalking, emerging vertically from the water and beating their tails to move backward as though walking on the water.
When Billie had recovered she was released back into the wild, but conservationist Mike Bossley continued to monitor her. He was surprised to see her tailwalking — she had never received any training during her recuperation and must have learned this spontaneously from the trained dolphins. “The behavior has no known utility in the wild and, as you can imagine, uses up a fair amount of energy, but Billie kept it up,” write biologists Hal Whitehead and Luke Rendell in The Cultural Lives of Whales and Dolphins (2014).
Not only that, but other wild dolphins began tailwalking themselves. Bossley reported, “Another female dolphin called Wave began performing the same behavior, but does so with much greater regularity than Billie. Four adult female dolphins have also been seen tailwalking,” as have several calves. Just last month conservationist Jenni Wyrsta saw a wild dolphin named Bianca do 33 tailwalks in a row in the Port River, in sets of two and three. “I’ve never seen a dolphin do double, let alone triple tailwalks,” she told the Portside Messenger.
Whitehead and Rendell write, “So, a behavior learned in captivity, with no obvious function beside play, has apparently gone on to become something of a hit in the wild and persists to this day, twenty-five years following Billie’s release and after Billie’s own death in 2009.”
Allan William Johnson discovered this “magic square of squares” in 1990:
302 2462 1722 452 932 1162 662 2582 1262 1382 2372 442 2602 32 542 1502
Each row, column, and long diagonal totals 93025, or 3052.
In the humble tetrahedron, each face shares an edge with each other face. Surprisingly, there’s only one other known polyhedron in which this is true — the Szilassi polyhedron, discovered in 1977 by Hungarian mathematician Lajos Szilassi:
If there’s a third such creature it would have 44 vertices and 66 edges, and no one knows whether such a shape could even be contrived. It remains an unsolved problem.
Arrange the first n2 odd numbers in a square (here n = 6):
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71
Now, no matter n, the sum of the first row is n2, the sum of either long diagonal is n3, and the sum of the whole array is n4.
(From Edward Barbeau’s Power Play, 1997.)
