Impression

This is remarkable — in 2014 ophthalmological neurologist Frederick Lepore showed this image to 100 migraine patients who experience a visual aura. (Click to enlarge.) 48 recognized it instantly.

“People are astonished,” Lepore told National Geographic. “They say, ‘Where did you get that?'”

He got it from English physician Hubert Airy, who had drawn his own aura in 1870, before the phenomenon was even understood.

(Via MetaFilter.)

Neat

The radius of a circle inscribed in a 3-4-5 triangle is 1.

(In fact, the inradius of any Pythagorean triangle is an integer.)

Theory and Practice

All new ventures have their detractors, and James had his full share with the Cavendish project. One diminishing but still powerful school of critics held that, while experiments were necessary in research, they brought no benefit to teaching. A typical member was Isaac Todhunter, the celebrated mathematical tutor, who argued that the only evidence a student needed of a scientific truth was the word of his teacher, who was ‘probably a clergyman of mature knowledge, recognised ability, and blameless character’. One afternoon James bumped into Todhunter on King’s Parade and invited him to pop into the Cavendish to see a demonstration of conical refraction. Horrified, Todhunter replied: ‘No, I have been teaching it all my life and don’t want my ideas upset by seeing it now!’

— Basil Mahon, The Life of James Clerk Maxwell, 2004

The Motorist’s Complaint

There are 8 ways to fold a 2 × 2 map along its creases:

But no one knows how to calculate the number of ways that a larger map might be folded. Amazingly, the number of ways to fold an n × n map has been worked out only as high as 7 × 7:

1, 8, 1368, 300608, 186086600, 123912532224, 129950723279272

Via Fermat’s Library.

The Cute Response

“Humans feel affection for animals with juvenile features,” noted Konrad Lorenz. “Large eyes, bulging craniums, retreating chins. Small-eyed, long-snouted animals do not elicit the same response.”

This induces people to care for small, cuddly animals. “And this has led some experts to argue that the entire phenomenon of pet-keeping is nothing more nor less than an elaborate case of social parasitism,” writes zoologist James Serpell. “Needless to say, this idea has done little to promote a positive view of pets or their owners. Rather, it creates the impression that pet-owners are the victims of some kind of bizarre affliction, and that dogs, cats and budgerigars are little different from body lice, fleas or tapeworms or, indeed, any other sort of parasitic organism.”

(From James Serpell, In the Company of Animals, 1986.)

Good Faith

“Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it is an experimental fact.” — Gabriel Lippmann

(Thanks, Tom.)

Nosedive

I just noticed this last night in Joseph Wood Krutch’s Treasury of Bird Lore — in 1832 ornithologist Alexander Wilson encountered a flock of passenger pigeons near Frankfort, Kentucky, that he estimated at 2,230,270,000 birds. If each bird ate only a pint of beech nuts in the course of a day, the flock would consume nearly 35 million bushels a day. A century and a half earlier, in 1687, Louis Armand, Baron de Lahontan, reported that pigeons had “so swarmed and ravaged the colonists’ crop near Montreal that a bishop was constrained to exorcise them with holy-water, as if they had been demons.”

Yet by 1914 human rapacity had reduced the species to a single bird, Martha, who died that year at the Cincinnati Zoo.

Slime Computing

In 2012 computer scientist Andrew Adamatzky set a plasmodium of the slime mold Physarum polycephalum into a maze with an oat flake at its center. By following a gradient of chemo-attractants given off by the flake, the plasmodium was able to solve the maze in one pass, extending a protoplasmic tube to the target.

The original recording was made at one frame per five minutes; this playback is 25-30 frames per second.

(Andrew Adamatzky, “Slime Mold Solves Maze in One Pass, Assisted by Gradient of Chemo-Attractants,” IEEE Transactions on Nanobioscience 11:2 [2012], 131-134.)

Thought

“It is interesting that most of the human race has a reserve of the enzyme necessary to render alcohol harmless to the body — as if nature meant us to drink alcohol, unlike animals to which alcohol is a poison.”

BUPA News, 1982, quoted in Richard Gordon, Great Medical Mysteries, 2014

Van der Waerden’s Theorem

Number eight cells:

Now suppose we want to color each cell red or blue such that no three cells are in arithmetic progression — for example, we don’t want cells 1, 2, and 3 to be the same color, or 4, 6, and 8. With eight cells it’s possible to accomplish this:

But if we want to add a ninth cell we can’t avoid an arithmetic progression: If the ninth cell is blue then cells 1, 5, and 9 are evenly spaced, and if it’s red then cells 3, 6, and 9 are. Dutch mathematician B.L. van der Waerden found that there’s always such a limit: For any given positive integers r and k, there’s some number N such that if the integers {1, 2, …, N} are colored, each with one of r different colors, then there will be at least k integers in arithmetic progression whose elements are of the same color. Determining what this limit is (in this example it’s 9) is an open problem.

(Bonus: Alexej Kanel-Belov found this pretty theorem concerning divisibility of integer sums within an infinite grid — Martin J. Erickson, in Beautiful Mathematics, calls it a two-dimensional version of van der Waerden’s theorem.)