# Math Notes

If a tetrahedron is constructed on a base with side lengths 125, 244, 267, then the remaining sides can take the shapes of three right triangles: (44, 240, 244), (44, 117, 125), (117, 240, 267).

And now the sum of the squares of the lengths of each pair of opposite sides in the tetrahedron is the same:

1172 + 2442 = 1252 + 2402 = 442 + 2672 = 73225

(From Edward Barbeau, Power Play, 1997.)

# First Sight

In 1728 English surgeon William Cheselden removed the cataracts from a 13-year-old boy, producing the first known case of full recovery from blindness:

Having often forgot which was the cat, and which the dog, he was ashamed to ask; but catching the cat, which he knew by feeling, he was observed to look at her steadfastly and then, setting her down said, ‘So, puss, I shall know you another time.’ He was very much surprised, that those things which he had liked best, did not appear most agreeable to his eyes, expecting those persons would appear to be most beautiful that he loved most, and such things to be most agreeable to his sight, that were so to his taste.

Also: “Being shewn his father’s picture in a locket at his mother’s watch, and told what it was, he acknowledged the likeness, but was vastly surprised; asking, how it could be, that a large face could be expressed in so little room, saying, it should have seemed as impossible for him, as to put a bushel of anything into a pint.” A fuller account is here. See also Molyneaux’s Problem.

# The Fibonacci Chimney

The smokestack of the local power plant in Turku, Finland, bears the first ten numbers of the Fibonacci sequence in glowing letters seven feet high.

The artist, Mario Merz, had been obsessed with the sequence for nearly 30 years when he added the numbers in 1995; he’d already added them to a chapel in Paris and a spire in Turin.

“It is entirely by accident that the sequence reflects two of the major research fields of the University of Turku, namely, number theory and mathematical biology,” write mathematicians Mats Gyllenberg and Karl Sigmund. “As is well known, Fibonacci introduced the sequence at around AD 1200 to model the growth of a rabbit population.”

(Mats Gyllenberg and Karl Sigmund, “The Fibonacci Chimney,” Mathematical Intelligencer 22:4 [2000], 46.)

# The Founder Effect

The population on the left has equal numbers of blue and red individuals. But if it colonized a new area using a very small number of individuals, one color or the other might predominate, with sometimes dramatic effects.

The Afrikaner population of South Africa is descended primarily from one shipload of immigrants that landed in 1652. One of these colonists carried the gene for Huntington’s disease, an autosomal dominant disease that causes a fatal breakdown of nerve cells in the brain. Most cases of the disease in the modern Afrikaner population can be traced to that individual.

Another condition, lipoid proteinosis, has been traced to Jacob Cloete, a German immigrant to the Cape in 1652. His great-grandson, Gerrit Cloete, migrated to Namaqualand in 1742. The area is somewhat isolated, so intermarriages were relatively common, compounding the effect.

(Thanks, Matt.)

# The Oxford Electric Bell

The University of Oxford has a bell that’s been ringing almost continuously since 1840. A little 4-millimeter clapper oscillates between two bells, each of which is positioned beneath a dry pile, an early battery. Due to the electrostatic force, the clapper is first attracted to and then repelled by each bell in turn, so it’s been ringing them alternately for 179 years. The operation conveys only a tiny amount of charge between the bells, which explains why it’s managed to run so long. The whole apparatus is kept under two layers of glass, but the ringing is so faint that it would be inaudible in any case.

It’s estimated that the bell has produced 10 billion rings so far — it holds the Guinness World Record as “the world’s most durable battery [delivering] ceaseless tintinnabulation.”

# In and Out

The briefest interview I’ve ever conducted was with Renato Dulbecco, who has since shared in a Nobel Prize for work in animal-cell culture and tumor viruses. Through his secretary, we had made an appointment. When I reached his office, he ushered me in, closed the door, sat down at his desk — and said that he was not going to talk to me. Startled, but respecting him at least for not having imposed on his secretary the task of rejection, I said something about the importance of getting scientific work across to the general public. Dulbecco replied, ‘We don’t do science for the general public. We do it for each other. Good day.’

— Horace Freeland Judson, “Reweaving the Web of Discovery,” The Sciences, November/December 1983

(“I thanked him for the interview and left, promising myself to use it someday. He was correct, of course, though unusually candid.”)

# Good Fortune

Letter from Albert Einstein to J.E. Switzer, April 23, 1953:

Dear Sir

Development of Western Science is based on two great achievements; the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationship by systematic experiment (Renaissance). In my opinion one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.

Sincerely yours,

A. Einstein

# Magicker

The “Lo Shu square” is the 3 × 3 square enclosed in dashed lines at the center of the diagram above. It’s “magic”: Each row, column, and long diagonal (marked in red) sums to 15. William Walkington has discovered a new magic property — imagine rolling the square into a tube (in either direction), and then bending the tube into a torus. And now imagine hopping from cell to cell around the torus with a “knight’s move” — two cells over and one up. (The extended diagram above helps with visualizing this — follow the blue lines.) It turns out that each such path touches three cells, and these cells always sum to 15. So the square is even more magic than we thought.

# Nose Harmony

English chemist Septimus Piesse likened scents to music:

Odors seem to affect the olfactory nerves in certain definite degrees, as sounds act on the auditory nerves. There is, so to speak, an octave of smells, as there is an octave of tones; some perfumes accord, like the notes of an instrument. Thus almond, vanilla, heliotrope, and clematis, harmonize perfectly, each of them producing almost the same impression in a different degree. On the other hand, we have citron, lemon, orange peel, and verbena, forming a similarly associated octave of odors, in a higher key. The analogy is completed by those odors which we call half-scents, such as the rose, with rose-geranium for its semitone; ‘petit-grain’ and neroli, followed by orange-flower. With the aid of flowers already known, by mixing them in fixed proportions, we can obtain the perfume of almost all flowers.

Using an “odaphone,” or scale, on which harmonies and discords of odors might be studied, his London perfumery Piesse and Lubin produced some of the most important scents of the Victorian era, such as Ambergris (1873), Hungary Water (1873), Kiss Me Quick (1873), The Flower of the Day (1875), White Rose (1875), and Frangipanni (1880).

It had been thought that none of these had survived, but in 2011 two unopened bottles were discovered in the bow of the Mary Celestia, a Civil War blockade runner that had foundered off Bermuda in 1864. The bottles contained Bouquet Opoponax, one of the company’s most popular fragrances, and after analysis with a gas chromatograph, Germany’s Drom Fragrances managed to reproduce the scent in 2014.

“I was shocked at how fresh and floral it was and by the amount of citrus in it,” senior perfumer Jean-Claude Delville told the Star-Ledger. “When the fragrance has been sitting at the bottom of the ocean and aging for so many years you expect something that is oxidized or damaged,” he told CTVNews. “But my first impression was ‘wow’.”

# The Citardauq Formula

This is interesting — if ax2 + bx + c = 0, then

$\displaystyle x = \frac{2c}{-b\mp \sqrt{b^{2} - 4ac}}.$

It’s called the citardauq formula — citardauq is quadratic spelled backward. It’s explained above by the inimitable James Tanton; here’s a Facebook discussion; and see the second answer here regarding questions of numerical instability.