Cutting Up

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Image: Wikimedia Commons

Choose any number of points on a circle and connect them to form a polygon.

This polygon can be carved into triangles in any number of ways by connecting its vertices.

No matter how this is done, the sum of the radii of the triangles’ inscribed circles is constant.

This is an example of a Sangaku (literally, “mathematical tablet”), a class of geometry theorems that were originally written on wooden tablets and hung as offerings on Buddhist temples and Shinto shrines during Japan’s Edo period (1603-1867). This one dates from about 1800.

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Chebyshev’s Paradoxical Mechanism

Russian mathematician Pafnuty Chebyshev devised this puzzling mechanisms in 1888. Turning the crank handle once will send the flywheel through two revolutions in the same direction, or four revolutions in the opposite direction. (A better video is here.)

“What is so unusual in this mechanism is the ability of the linkages to flip from one configuration to the other,” write John Bryant and Chris Sangwin in How Round Is Your Circle? (2011). “In most linkage mechanisms such ambiguity is implicitly, or explicitly, designed out so that only one choice for the mathematical solution can give a physical configuration. … This mechanism is really worth constructing, if only to confound your friends and colleagues.”

(Thanks, Dre.)

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All-Stars

1927 solvay conference

At the Fifth Solvay International Conference, held in Brussels in October 1927, 29 physicists gathered for a group photograph. Back row: Auguste Piccard, Émile Henriot, Paul Ehrenfest, Édouard Herzen, Théophile de Donder, Erwin Schrödinger, Jules-Émile Verschaffelt, Wolfgang Pauli, Werner Heisenberg, Ralph Howard Fowler, Léon Brillouin. Middle: Peter Debye, Martin Knudsen, William Lawrence Bragg, Hendrik Anthony Kramers, Paul Dirac, Arthur Compton, Louis de Broglie, Max Born, Niels Bohr. Front: Irving Langmuir, Max Planck, Marie Sklodowska Curie, Hendrik Lorentz, Albert Einstein, Paul Langevin, Charles-Eugène Guye, Charles Thomson Rees Wilson, Owen Willans Richardson.

Seventeen of the 29 were or became Nobel Prize winners. Marie Curie, the only woman, is also the only person who has won the prize in two scientific disciplines.

Below: On Aug. 12, 1958, 57 notable jazz musicians assembled for a group portrait at 17 East 126th Street in Harlem. They included Red Allen, Buster Bailey, Count Basie, Emmett Berry, Art Blakey, Lawrence Brown, Scoville Browne, Buck Clayton, Bill Crump, Vic Dickenson, Roy Eldridge, Art Farmer, Bud Freeman, Dizzy Gillespie, Tyree Glenn, Benny Golson, Sonny Greer, Johnny Griffin, Gigi Gryce, Coleman Hawkins, J.C. Heard, Jay C. Higginbotham, Milt Hinton, Chubby Jackson, Hilton Jefferson, Osie Johnson, Hank Jones, Jo Jones, Jimmy Jones, Taft Jordan, Max Kaminsky, Gene Krupa, Eddie Locke, Marian McPartland, Charles Mingus, Miff Mole, Thelonious Monk, Gerry Mulligan, Oscar Pettiford, Rudy Powell, Luckey Roberts, Sonny Rollins, Jimmy Rushing, Pee Wee Russell, Sahib Shihab, Horace Silver, Zutty Singleton, Stuff Smith, Rex Stewart, Maxine Sullivan, Joe Thomas, Wilbur Ware, Dickie Wells, George Wettling, Ernie Wilkins, Mary Lou Williams, and Lester Young. Photographer Art Kane called it “the greatest picture of that era of musicians ever taken.”

http://www.wikiwand.com/en/A_Great_Day_in_Harlem_(photograph)

| Art · Science & Math

The Wisdom of the Crowd

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At a livestock exhibition at Plymouth, England, in 1907, attendees were invited to guess the weight of an ox and to write their estimates on cards, with the most accurate estimates receiving prizes. About 800 tickets were issued, and after the contest these made their way to Francis Galton, who found them “excellent material.”

“The average competitor,” he wrote, “was probably as well fitted for making a just estimate of the dressed weight of the ox, as an average voter is of judging the merits of most political issues on which he votes, and the variety among the voters to judge justly was probably much the same in either case.”

Happily for all of us, he found that the guesses in the aggregate were quite accurate. The middlemost estimate was 1,207 pounds, and the weight of the dressed ox proved to be 1,198 pounds, an error of 0.8 percent. This has been borne out in subsequent research: When a group of people make individual estimates of a quantity, the mean response tends to be fairly accurate, particularly when the crowd is diverse and the judgments are independent.

Galton wrote, “This result is, I think, more creditable to the trustworthiness of a democratic judgment than might have been expected.”

(Francis Galton, “Vox Populi,” Nature, March 7, 1907.)

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Nicomachus’ Theorem

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Image: Wikimedia Commons

In 100 C.E., Nicomachus of Gerasa observed that

13 + 23 + 33 + … + n3 = (1 + 2 + 3 + … + n)2

Or “the sum of the cubes of 1 to n is the same as the square of their sum.” The diagram above demonstrates this neatly: Counting the individual squares shows that

1 × 12 + 2 × 22 + 3 × 32 + 4 × 42 + 5 × 52 + 6 × 62
= 13 + 23 + 33 + 43 + 53 + 63
= (1 + 2 + 3 + 4 + 5 + 6)2

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Math Notes

From a 1951 issue of The Dark Horse, the staff magazine of Lloyds Bank, a bitter mnemonic for pi:

Now I live a drear existence in ragged suits
And cruel taxation suffering.

3.141592653589

Also, a curiosity:

(3,1,4) = (1,5,9) + (2,6,5) (mod 10)

(Thanks, Trevor.)

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The Butterfly Theorem

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Image: Wikimedia Commons

Draw a circle, choose any chord PQ, and draw two further chords AB and CD through its midpoint M. Now, if AD and BC intersect PQ at X and Y, M will always be the midpoint of XY.

In Icons of Mathematics (2011), Claudi Alsina and Roger Nelsen write, “The surprise is the unexpected symmetry arising from an almost random construction.” The theorem first appeared in 1815.

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Averageness

galton criminal composites

In 1883 Francis Galton tried an experiment: He combined multiple photographs of criminals into composite images, hoping to discover an underlying “type.” He didn’t get a strong result, but he did notice something odd about the composite faces: They tended to be more attractive than the individual images that made them up. He found similar effects with other groups — a composite “sick person” seemed healthier than its constituent images, and a group of good-looking people became even more beautiful in composite. In one case he made a “singularly beautiful combination of the faces of six different Roman ladies, forming a charming ideal profile.”

The lesson seems to be that we find an “average” face most attractive — a face is appealing not because it has unusual features but because it lacks them. For example (below), a University of Toronto study found that the shape of Jessica Alba’s face approaches the average for all female profiles: The distance between her pupils is 46 percent of the width of her face, and the distance between her eyes and her mouth is 36 percent of the length of her face. The fact that we find this attractive makes some evolutionary sense: Natural selection tends to drive out disadvantageous features, so a partner with an “average” face is more likely to be healthy and fertile.

https://commons.wikimedia.org/wiki/File:Jessica_Alba_Face_Proportions.png
Image: Wikimedia Commons
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