Draw a triangle, pick a point on one side, and draw a path as shown, with each segment parallel to a side of the triangle.
The path with always be closed — you’ll always return to your starting point.
Discovered by German mathematician Gerhard Thomsen.
I just ran across this anecdote by Jason Rosenhouse in Notices of the American Mathematical Society. In a middle-school algebra class Rosenhouse’s brother was given this problem:
There are some horses and chickens in a barn, fifty animals in all. Horses have four legs while chickens have two. If there are 130 legs in the barn, then how many horses and how many chickens are there?
The normal solution is straightforward, but Rosenhouse’s brother found an alternative that’s even easier: “You just tell the horses to stand on their hind legs. Now there are fifty animals each with two legs on the ground, accounting for one hundred legs. That means there are thirty legs in the air. Since every horse has two legs in the air, we find that there are fifteen horses, and therefore thirty-five chickens.”
(Jason Rosenhouse, “Book Review: Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles,” Notices of the American Mathematical Society, 67:9 [October 2020], 1382-1385.)
Grab point C above and drag it to a new location. Surprisingly, M, the midpoint of BaAb, doesn’t move.
This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.
It was discovered by Dutch mathematician Oene Bottema.
In 1898 a Belgian ship on a scientific expedition was frozen into the sea off the coast of Antarctica. During the long polar night, its 18 men would confront fear, death, illness, and despair. In this week’s episode of the Futility Closet podcast we’ll describe life aboard the Belgica during its long, dark southern winter.
We’ll also consider a devaluing signature and puzzle over some missing music.
In 1963, French geologist Michel Siffre descended into a glaciated cavern under the French-Italian Maritime Alps and spent more than two months without sun to “investigate time, that most inapprehensible and irreversible thing.” He could telephone scientists on the surface, who recorded the time of each call, but they never told him the time or date. From his diary:
Forty-second awakening. … I really seem to have no least idea of the passage of time. This morning, as an example, after telephoning to the surface and talking for a while, I wondered afterward how long the telephone conversation had lasted, and could not even hazard a guess. … Fifty-second awakening. … I am losing all notion of time. … When, for instance, I telephone the surface and indicate what time I think it is, thinking that only an hour has elapsed between my waking up and eating breakfast, it may well be that four or five hours have elapsed. And here is something hard to explain: the main thing, I believe, is the idea of time that I have at the very moment of telephoning. If I called an hour earlier, I would still have stated the same figure. … I am having great difficulty to recall what I have done today. It costs me a real intellectual effort to recall such things.
The outsiders could see that his waking and sleeping remained near a cycle of 24.5 hours, but Siffre’s conscious understanding of time was greatly affected. Misunderstanding the length of his day, he began to husband his rations, thinking he had weeks more to endure. At his 57th awakening, the final day of the experiment, he thought it was August 20; in fact it was September 14. “I underestimated by almost half the length of my working or waking hours; a ‘day’ that I estimated at seven hours actually lasted on the average fourteen hours and forty minutes.” NASA has pursued these inquiries to consider the implications for space travelers.
(From Jane Brox, Brilliant: The Evolution of Artificial Light, 2010.)
This is interesting: USC urban planning professor Geoff Boeing examined the street networks of 100 world cities as a measure of their spatial logic and order.
The cities with the most ordered streets are Chicago, Miami, and Minneapolis; most disordered are Charlotte, São Paulo, and Rome.
“On average, US/Canadian study sites are far more grid-like than those elsewhere, exhibiting less entropy and circuity.”
(Geoff Boeing, “Urban Spatial Order: Street Network Orientation, Configuration, and Entropy,” Applied Network Science 4:1 [2019], 1-19.) (Via Ethan Mollick.)
For more than 40 years in the early 20th century, Martin Couney ran a sideshow in which premature babies were displayed in incubators. With this odd practice he offered a valuable service in an era when many hospitals couldn’t. In this week’s episode of the Futility Closet podcast we’ll describe Couney’s unusual enterprise, which earned both criticism and praise.
We’ll also marvel over an Amazonian survival and puzzle over a pleasing refusal.
How does an outfielder know where to run in order to catch a fly ball? Previously it had been thought that the fielder estimates the ball’s arc, acceleration, and distance; predicts where it will land; and runs straight to that spot.
“That was a really elegant solution,” Kent State psychologist Michael McBeath told the New York Times in 1995. “The only problem is that keeping track of acceleration like that is something that people are very bad at.”
McBeath and his colleagues analyzed fly balls and catches visually, mathematically, and subjectively from the players’ perspective, using a video camera. They found that fielders learn to run so that the ball follows a straight line in their visual field. “If you are faster than the critter you are trying to catch, if you can keep the prey on a simple path in your vision — hold it as if it’s moving in a straight line in your eye — then you’ll catch it.”
Among other things, this explains why fielders sometimes collide with walls when chasing uncatchable home runs. They haven’t calculated in advance where the ball will come down; instead they’re following an algorithm that’s directing them, accurately, to a landing point that’s not on the field.
(Michael K. McBeath, Dennis M. Shaffer, and Mary K. Kaiser, “How Baseball Outfielders Determine Where to Run to Catch Fly Balls,” Science 268:5210 [1995], 569-573.) (See Shortcuts.)
Jakob von Uexküll used to say: ‘When a dog runs, the dog moves its legs; when a sea urchin runs, the legs move the sea urchin.’ This assertion was based on the following experiment reported by von Uexküll. A sea urchin was broken in half and the inner sides of both halves of the shell were scraped using sandpaper. The whole of the ambulacral system as well as the nervous system was thus completely removed. Then the two halves were joined together again by means of a spring clasp. The spines of the sea urchin still worked in coordination with one another. In this special case, the riderless horse of von Holst’s parable does indeed exist; the sea urchin’s reaction of fleeing from a star fish still functioned. And in this sense, von Uexkull’s description of a sea urchin being a ‘reflex republic’ is justified, provided one keeps in mind that the ‘reflex’ no longer plays the all-important role ascribed to it during von Uexküll’s time.
— Konrad Lorenz, The Foundations of Ethology, 1982
(Lorenz described this direct mutual influence among the peripheral organs as “a panic spreading among the spines.”)
ONE + TWO – THREE – FOUR + FIVE = 1
That’s true if we replace each word either with the number it denotes or with the quantity of its letters: Either way we’re left with 1. Another:
ONE + TWO – THREE – FOUR + FIVE – SIX + SEVEN + EIGHT + NINE – TEN + ELEVEN + TWELVE – THIRTEEN – FOURTEEN = 5
These are the only two such sequences using 20 or fewer consecutive number names, found Leonard Gordon, although other sequences of plus and minus signs are possible.
In a separate but related project, Gordon assigned the number names ONE through FIFTEEN, ONE through NINETEEN, and ONE through TWENTY to either side of an equals sign so that the denoted equation is mathematically correct and each equation “balances,” with the same number of letters on each side:
ONE + FOUR + SEVEN + TEN + ELEVEN + THIRTEEN + FOURTEEN = TWO + THREE + FIVE + SIX + EIGHT + NINE + TWELVE + FIFTEEN
ONE + THREE + FIVE + SEVEN + NINE + SIXTEEN + SEVENTEEN + EIGHTEEN + NINETEEN = TWO + FOUR + SIX + EIGHT + TEN + ELEVEN + TWELVE + THIRTEEN + FOURTEEN + FIFTEEN
ONE + THREE + SIX + NINE + TEN + TWELVE + THIRTEEN + FIFTEEN + SEVENTEEN + NINETEEN = TWO + FOUR + FIVE + SEVEN + EIGHT + ELEVEN + FOURTEEN + SIXTEEN + EIGHTEEN + TWENTY
(“Self-Referential Sums Revisited,” in “Kickshaws,” Word Ways 28:3 [August 1995], 170-180.)
