Arithmetic Billiards

https://commons.wikimedia.org/wiki/File:Arithmetic-billiard-40-15.jpg

To find the least common multiple and the greatest common divisor of two natural numbers, construct a billiard table whose side lengths correspond to the two numbers (here, 15 and 40). Set a ball in one corner, fire it out at a 45-degree angle, and let it bounce around the table until it stops in a corner.

Now the least common multiple is the total number of unit squares traversed by the ball (here, 120).

And the greatest common divisor is the number of unit squares traversed by the ball before it reaches the first intersection (here, 5).

More details here.

| Science & Math

A Prime Formula

A team of mathematicians in Canada and Japan discovered this remarkable polynomial in 1976 — let its 26 variables a, b, c, … z range over the non-negative integers and it will generate all prime numbers:

\displaystyle (k+2)(1-\newline [wz+h+j-q]^{2}-\newline [(gk+2g+k+1)(h+j)+h-z]^{2}-\newline [16(k+1)^{3}(k+2)(n+1)^{2}+1-f^{2}]^{2}-\newline [2n+p+q+z-e]^{2}-\newline [e^{3}(e+2)(a+1)^{2}+1-o^{2}]^{2}-\newline [(a^{2}-1)y^{2}+1-x^{2}]^{2}-\newline [16r^{2}y^{4}(a^{2}-1)+1-u^{2}]^{2}-\newline [n+\ell +v-y]^{2}-\newline [(a^{2}-1)\ell ^{2}+1-m^{2}]^{2}-\newline [ai+k+1-\ell -i]^{2}-\newline [((a+u^{2}(u^{2}-a))^{2}-1)(n+4dy)^{2}+1-(x+cu)^{2}]^{2}-\newline [p+\ell (a-n-1)+b(2an+2a-n^{2}-2n-2)-m]^{2}-\newline [q+y(a-p-1)+s(2ap+2a-p^{2}-2p-2)-x]^{2}-\newline [z+p\ell (a-p)+t(2ap-p^{2}-1)-pm]^{2})\newline >0

The snag is that it will sometimes produce negative numbers, which must be ignored. But every positive result will be prime, and every prime can be generated by some set of 26 non-negative integers.

(James P. Jones et al., “Diophantine Representation of the Set of Prime Numbers,” American Mathematical Monthly 83:6 [1976], 449-464.)

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Spreading Word

When a Western scrub jay discovers the body of a dead jay, it summons other birds to screech over the body for up to half an hour. It’s not clear why they do this — the birds are territorial and not normally social. Possibly it’s a way to share news of danger, concentrate attention to find a predator, or teach young about dangers in the environment.

The gatherings are sometimes called funerals, though we don’t know enough to understand the reasons behind them. But UC Davis student Teresa Iglesias said, “I think there’s a huge possibility that there is much more to learn about the social and emotional lives of birds.”

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Something Else

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Image: Wikimedia Commons

During a visit to Oxford in May 1931, Albert Einstein gave a brief lecture on cosmology, and afterward the blackboard was preserved along with Einstein’s ephemeral writing. It now resides in the university’s Museum of the History of Science.

Harvard historian of science Jean-François Gauvin argues that this makes it a “mutant object”: It’s no longer fulfilling the essential function of a blackboard, to store information temporarily — it’s become something else, a socially created object linked to the great scientist. The board’s original essence could be restored by wiping it clean, but that would destroy its current identity.

“The sociological metamorphosis at the origin of this celebrated artifact has completely destroyed its intrinsic nature,” Gauvin writes. “Einstein’s blackboard has become an object of memory, an object of collection modified at the ontological level by a social desire to celebrate the achievement of a great man.”

| Oddities · Science & Math

Wishful Thinking

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In the 1850s, when a plague of spirits began to rotate tables at London séances, Michael Faraday devised a clever way to investigate: Two boards were lain one atop the other, with an upright haystalk inserted through the pair. The experimenters laid their hands on this. The apparatus gave a way to see “whether the table moved the hand, or the hand moved the table”: If the medium willed the table to move to the left, and it did so on its own, the haystalk would be seen to lean in one direction … but if the experimenters, even unconsciously, themselves pressed the table to turn, it would lean in the other.

Faraday wrote, “As soon as the index is placed before the most earnest, and they perceive — as in my presence they have always done — that it tells truly whether they are pressing downwards only or obliquely, then all effects of table-turning cease, even though the parties persevere, earnestly desiring motion, till they become weary and worn out. No prompting or checking of the hands is needed — the power is gone; and this only because the parties are made conscious of what they are really doing mechanically, and so are unable unwittingly to deceive themselves.”

(Michael Faraday, “On Table-Turning,” Times, June 30, 1853.)

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Careful

Richard Feynman tangled regularly with military censors at Los Alamos. Playing one day with a computing machine, he discovered a pleasing little pattern:

1/243 = 0.004115226337448559670781893004115226337448559670781893004115226…

“It’s quite cute, and then it goes a little cockeyed when you’re carrying; confusion occurs for only about three numbers, and then you can see how the 10 10 13 is really equivalent to 114 again, or 115 again, and it keeps on going, and repeats itself nicely after a couple of cycles. I thought it was kind of amusing.”

Well, I put that in the mail, and it comes back to me. It doesn’t go through, and there’s a little note: ‘Look at Paragraph 17B.’ I look at Paragraph 17B. It says, ‘Letters are to be written only in English, Russian, Spanish, Portuguese, Latin, German, and so forth. Permission to use any other language must be obtained in writing.’ And then it said, ‘No codes.’

So I wrote back to the censor a little note included in my letter which said that I feel that of course this cannot be a code, because if you actually do divide 1 by 243 you do, in fact, get all that, and therefore there’s no more information in the number .004115226337… than there is in the number 243 — which is hardly any information at all. And so forth.

“I therefore asked for permission to use Arabic numerals in my letters. So, I got that through all right.”

(From his reminiscences.)

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Harmony

In 1995, Alma College mathematician John F. Putz counted the measures in Mozart’s piano sonatas, comparing the length of the exposition (a) to that of the development and recapitulation (b):

Köchel and movement a b a + b
279, I 38 62 100
279, II 28 46 74
279, III 56 102 158
280, I 56 88 144
280, II 56 88 144
280, II 24 36 60
280, III 77 113 190
281, I 40 69 109
281, II 46 60 106
282, I 15 18 33
282, III 39 63 102
283, I 53 67 120
283, II 14 23 37
283, III 102 171 273
284, I 51 76 127
309, I 58 97 155
311, I 39 73 112
310, I 49 84 133
330, I 58 92 150
330, III 68 103 171
332, I 93 136 229
332, III 90 155 245
333, I 63 102 165
333, II 31 50 81
457, I 74 93 167
533, I 102 137 239
533, II 46 76 122
545, I 28 45 73
547, I 78 118 196
570, I 79 130 209

He found that the ratio of b to a + b tends to match the golden ratio. For example, the first movement of the first sonata is 100 measures long, and of this the development and recapitulation make up 62. “This is a perfect division according to the golden section in the following sense: A 100-measure movement could not be divided any closer (in natural numbers) to the golden section than 38 and 62.”

Ideally there are two ratios that we could hope would hew to the golden section: The first relates the number of measures in the development and recapitulation section to the total number of measures in each movement, and the second relates the length of the exposition to that of the recapitulation and development. The first of these gives a correlation coefficient of 0.99, the second of only 0.938.

So it’s not as impressive as it might be, but it’s still striking. “Perhaps the golden section does, indeed, represent the most pleasing proportion, and perhaps Mozart, through his consummate sense of form, gravitated to it as the perfect balance between extremes,” Putz writes. “It is a romantic thought.”

(John F. Putz, “The Golden Section and the Piano Sonatas of Mozart,” Mathematics Magazine 68:4 [October 1995], 275-282.)

| Art · Science & Math

Climate Music

In 2013, University of Minnesota geography student Daniel Crawford composed “A Song of Our Warming Planet,” a solo cello piece built on climate data. The pitch of each note corresponds to the average annual surface temperature of a year in the range 1880-2012 in data from NASA’s Goddard Institute of Space Studies; each ascending halftone represents roughly 0.03°C in planetary warming.

“Climate scientists have a standard toolbox to communicate their data,” Crawford said. “We’re trying to add another tool to that toolbox, another way to communicate these ideas to people who might get more out of music than maps, graphs, and numbers.”

Below: He later applied the same method to create a string quartet using data from 1880 to 2014. The four parts reflect the average annual temperatures in four regions: the cello the equatorial zone, the viola the mid-latitudes, and the violins the high latitudes and the arctic.

| Art · Science & Math

A Sleeping Cat

Here’s a surprise: A new geoglyph has been found in the soil of the Nazca Desert in southern Peru.

This one, evidently a cat, appears to be the oldest yet — it may have been engraved in the earth as early as 200 B.C. and has been waiting all this time to be discovered.

“The figure was barely visible and was about to disappear because it is situated on quite a steep slope that’s prone to the effects of natural erosion,” the culture ministry wrote.

“It’s quite striking that we’re still finding new figures,” said chief archaeologist Johny Isla, “but we also know that there are more to be found.”

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