Curve Stitching

http://en.wikipedia.org/wiki/File:Quadratic_Beziers_in_string_art.svg
Image: Wikimedia Commons

Mary Everest Boole, the wife of logician George Boole, was an accomplished mathematician in her own right. In order to convey mathematical ideas to young people she invented “curve stitching,” the practice of constructing straight-line envelopes by stitching colored thread through a pattern of holes pricked in cardboard. In each of the examples above, two straight lines are punctuated with holes at equal intervals, defining a quadratic Bézier curve. When the holes are connected with thread as shown, their envelope traces a segment of a parabola.

“Once the fundamental idea of the method has been mastered, anyone interested can construct his own designs,” writes Martyn Cundy in Mathematical Models (1952). “Exact algebraic curves will usually need unequal spacing of the holes and therefore more calculation will be required to produce them; it is surprising, however, what a variety of beautiful figures can be executed which are based on the simple principle of equal spacing.”

The American Mathematical Society has some patterns and resources.

| Science & Math

Math and Poetry

In 1972 the Belgian mathematician Edouard Zeckendorf established Zeckendorf’s theorem: that every positive integer can be represented as the sum of non-consecutive Fibonacci numbers in one and only one way.

In 1979 French poet Paul Braffort celebrated this with a series of 20 poems, My Hypertropes. Each of the 20 poems in the series is informed by the foregoing poems that make up its Zeckendorff sum. For example, the Zeckendorff representation of 12 is 8 + 3 + 1, so poem 12 in Braffort’s sequence shares some characters or images with each of these poems. This forced Braffort to build scenarios that would permit these relations as he wrote the poems.

Each of the numbers 1, 2, 3, 5, 8, and 13 is its own Zeckendorff representation, so Braffort related each of these to its two foregoing Fibonacci numbers (e.g., 8 = 3 + 5). This means that only the first poem, “The Preallable Explanation (or The Rhyme’s Reason),” is not influenced by any of the others. Here is that first poem, as translated by Amaranth Borsuk and Gabriela Jaurequi:

This is my work, this is my study,
like Jarry, Cyrano puffy,

to split hairs on Rimbaud
and on willies find booboos.

If it was fair or if it snowed
in Lhassa Emma Sophie Bo-

vary widow of slow carnac
gave herself to the god of wack.

Leibnitz, saying: “Verse …” What an ac-
tor for this superb “Vers …”. Oh “nach”!

He aims, Emma, the apoplexy
of those drunk on galaxy.

At the club of “spinach” kings (nay,
Bach never went there, Banach yea!)

Leibnitz — his graph ibo: not six
mus, three nus, one phi, bona xi —

haunts without profit Bonn: “Ach! Gee
if I were great Fibonacci!!! …”

Now, for example, Poem 12, “MODELS (for Petrovich’s Band),” is an alexandrine with two six-line stanzas. The Zeckendorff representation of 12 is 1 + 3 + 8, so in each stanza of Poem 12 the first line is influenced by Poem 1, the third by Poem 3, and the sixth by Poem 8, each drawing on specific lines in the source poem. The first line in the sixth couplet of Poem 1, “He aims, Emma, the apoplexy,” informs the first line of Poem 12, “For a sweet word from Emma: a word for model”; the second line of the sixth couplet from Poem 1, “of those drunk on galaxy,” informs the first line of the second stanza in Poem 12, “Our galaxies have already packed their valise”; the phrase “when I saw you / weave a letter to Elise” in Poem 3 becomes “they say from this time forth five letters to Elise” in Poem 12; and the couplet “And Muses who compose / They’re a troop they’re tropes” in Poem 8 becomes “Tragic tropes: Leonardo is Fibonacci.”

“Thus, Braffort’s collection of poems, My Hypertropes, has an internal structure provided by a mathematical theorem,” writes Robert Tubbs in Mathematics in Twentieth-Century Literature and Art (2014). “The structure does not entirely determine these poems, but it does provide connections between the poems that might not be there otherwise.”

| Poems · Science & Math

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.)

| Oddities · Science & Math · Technology

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.)

| Science & Math · Society

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

| Science & Math

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.)

| Science & Math

The Butterfly Theorem

https://commons.wikimedia.org/wiki/File:Butterfly_theorem.svg
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.

| Science & Math