Cultivation

Many people know that you can form a pentagon by tying a strip of paper in a simple overhand knot.

Stephen Bleecker Luce’s seamanship manual of 1863 tells how to fold a blade of grass into an octagon (below):

It is first doubled short over itself, then 1 under 2, — leaving a space, then 2 over 1, and down through the centre of the triangle; next 1 over 2, and down through the centre, coming out on the opposite side, and so on until an octagonal figure is formed.

That’s pretty terse, but I think I’ve almost managed to do it tonight. Keep your eye on the drawing of the finished piece, and don’t form and flatten the finished shape until you’ve done all the weaving.

(Via The Ashley Book of Knots.)

Math Notes

The sum or difference of any pair of the numbers {150568, 420968, 434657} is a square:

420968 + 150568 = 7562
420968 – 150568 = 5202
434657 + 420968 = 9252
434657 – 420968 = 1172
434657 + 150568 = 7652
434657 – 150568 = 5332

Miracles and Agents

Jones tells a mountain to hop into the sea and it does so. Has he performed a miracle?

Well, no, writes University of Birmingham philosopher George Chryssides. If Jones repeats his feat, then he’s revealed an underlying causal principle that’s amenable to study just like the rest of the natural world. If he doesn’t repeat the feat, then there’s no support for the idea of a link between his command and the mountain’s movement — we know only that the two events coincided, not that one caused the other.

“In order … to determine the answer to the question, ‘Did Jones move the mountain?’ … we must ascertain whether similar effects would follow similar putative causes,” Chryssides writes. “Either an allegedly miraculous event is a violation of scientific law, in which case it could not be performed by an agent, or else it is performed by an agent, in which case it could not be a violation of scientific law.”

(George D. Chryssides, “Miracles and Agents,” Religious Studies 11:3 [September 1975], 319-327.)

Levine’s Sequence

In the summer of 1997 mathematician Lionel Levine discovered a sequence of numbers: 1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, … . It’s made up of the final term in each row of this array:

1 1
1 2
1 1 2
1 1 2 3
1 1 1 2 2 3 4
1 1 1 1 2 2 2 3 3 4 4 5 6 7
1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 9 10 10 11 12 13 14

The array is built from a simple rule. Start with 1 1 and regard each line as a recipe for building the next one: Read it from right to left and think of it as an inventory of digits. The first line, 1 1, would be read “one 1 and one 2,” so that gives us 1 2 for the second line. The second line (again, reading from right to left) would be read “two 1s and one 2,” giving 1 1 2 for the third line.

And so on. That’s it. It’s excruciatingly simple, “yet it seems likely that the 20th term say is impossible to compute,” wrote N.J.A. Sloane that November. At that time only 15 terms were known, the last being 508009471379488821444261986503540. Two further terms have since been found, bringing us up to 5347426383812697233786139576220450142250373277499130252554080838158299886992660750432. And the 20th is still out of sight.

See the link below (page 15) for more.

(Neil J.A. Sloane, “My Favorite Integer Sequences,” in C. Ding, T. Helleseth, and H. Niederreiter, Sequences and Their Applications: Proceedings of SETA ’98, 2012, 103-130.)

Words and Music

In 2004, Canadian musician Andrew Huang wrote a song that encodes the first 101 digits of π.

Also: A “piku” is a haiku whose word lengths reflect the digits of π:

How I love a verse
Contrived to unhusk dryly
One image nutshell

Zipf’s Law

In natural language, the most frequent word occurs about twice as often as the second most frequent word, three times as often as the third most frequent word, and so on.

In the Brown Corpus, a text collection of a million words, the most frequent word, the, accounts for 7.5% of all word occurrences, and the second most frequent, of, accounts for 3.5%. A mere 135 vocabulary items account for half the corpus, and about half the total vocabulary of about 50,000 words are hapax legomena, words that occur once only.

Similar distributions are found in data throughout the physical and social sciences; the law is named after the American linguist George Kingsley Zipf.

Cover Story

A set of points has diameter 1 if no two points in the set are more than 1 unit apart. An example is an equilateral triangle whose side has length 1. What’s the smallest shape that can cover any such set? A circle of diameter 1 won’t cover our triangle; part of the triangle projects beyond the circle:

Of course a larger circle would work, but what’s the smallest shape will always do the job? Surprisingly, no one knows. When French mathematician Henri Lebesgue posed the problem to Gyula Pál in 1914, Pál suggested a modified hexagon (in black):

Here Pál’s shape manages to surround a circle (blue), a Reuleaux triangle (red), and a square (green), each of diameter 1, and in fact it will accommodate any such set. Its own area is 0.84529946. Will a smaller shape do the job? Well, yes, but the gains get increasingly fine: In 1936 Roland Sprague whittled Pál’s shape down to 0.844137708436, and in 1992 H.C. Hansen reduced it further to 0.844137708398. At this point observers Victor Klee and Stanley Wagon wrote, “[I]t does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.” But in 2015 John Baez reached 0.8441153 with an exquisite adjustment to two regions in Hansen’s shape; the smaller of these would span only a few atoms if the shape were drawn on paper.

Is that the end of the story? No: Last October Philip Gibbs claimed a further reduction to 0.8440935944, and the search goes on. In 2005 Peter Brass and Mehrbod Sharifi showed that the universal cover must have an area of at least 0.832, so there’s room, at least in theory, for still further improvements.

(Thanks, Jacob.)

Spine Tinglers

In a 2009 study of responses to music, neuroscientist Valorie Salimpoor and her colleagues asked participants to bring in 3 to 5 pieces of “intensely pleasurable instrumental music to which they experience chills.” Then they measured their physiological response as they listened. They found that the “chills” effect is real — when the subjects reported that their pleasure at the music was highest, so was their sympathetic nervous system activity, a measure of emotional arousal.

One byproduct of the study is a list of more than 200 chills-inducing moments in music of various genres, with precise timestamps of the crucial points:

 Composer/Artist Title Chills Beethoven Piano Sonata No. 17 in D Minor (“The Tempest”) 5:33 Mahler Symphony No. 1 – Movement 4 5:42, 9:57, 15:15 Charles Mingus Fables of Faubus 0:20, 7:10 Stan Getz Round Midnight 1:26 Pink Floyd Shine on You Crazy Diamond 5:00 Phish You Enjoy Myself 10:50 Cannonball Adderley One for Daddy-O 0:40 Los Angeles Guitar Quartet Congan 2:09 Crowfoot Larks in May 0:10, 2:00 Howard Shore The Breaking of the Fellowship (film score) 0:10, 0:55 Dave Matthews Band #34 1:40 The Dissociatives Paris Circa 2007 Slash 08 1:30 Brad Mehldau Knives Out 4:45, 7:25 Explosions in the Sky First Breath After Coma 2:25, 3:30, 8:10

These won’t work for everyone — music tastes are notoriously idiosyncratic — but it’s interesting to see what people find moving. The full list is here (Table_S1). (Note too that the timestamps relate to a particular recording, so consider them approximate in e.g. classical music.)

(Valorie N. Salimpoor, et al., “The Rewarding Aspects of Music Listening Are Related to Degree of Emotional Arousal,” PloS One 4:10 [2009], e7487.)

Practice

While working on his chemistry doctorate in 1947, Isaac Asimov was dissolving catechol in water when it occurred to him that if it were any more soluble it would dissolve before it even touched the surface. Amused by the idea, he invented a fictional substance called thiotimoline, one of whose chemical bonds projects forward into the future and another backward into the past. This makes the chemical “endochronic”: It starts dissolving before it makes contact with water. His first thought was to make this into a science fiction story.

It occurred to me, however, that instead of writing an actual story based on the idea, I might write up a fake research paper on the subject and get a little practice in turgid writing. I did the job on June 8, 1947, even giving it the kind of long-winded title that research papers so often have — ‘The Endochronic Properties of Resublimated Thiotimoline’ — and added tables, graphs, and fake references to non-existent journals.

John W. Campbell of Astounding Science Fiction accepted the article and agreed to publish it under a pseudonym, lest it alienate Asimov’s examiners at Columbia. In the end he published it under Asimov’s own name, but there was no harm done — the examiners joked about it at his defense and it even brought him some fame among chemists. He went on to write three short stories about the substance — which has taken on a rich existence in the hands of other authors.

(Isaac Asimov, “The Endochronic Properties of Resublimated Thiotimoline,” Astounding Science Fiction 41:1 [1948], 120-125. Thanks, Peter.)

Grice’s Maxims

What rules underlie natural conversation? In a lecture at Harvard in 1967, British philosopher H.P. Grice set out to specify them using a mathematical approach, as Euclid had done in plane geometry. First, he said, the participants in a conversation follow a Cooperative Principle:

Make your conversational contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged.

Then he derived more specific principles under four headings:

• Quantity
1. Make your contribution as informative as is required.
• Quality
1. Try to make your contribution one that is true.
2. Do not say what you believe to be false.
3. Do not say that for which you lack adequate evidence.
• Relation
1. Be relevant.
• Manner
1. Be perspicuous.
2. Avoid obscurity of expression.
3. Avoid ambiguity.
4. Be brief.
5. Be orderly.

These are useful, but they’re not axioms. “[I]t is possible to engage in a genuine and meaningful conversation and yet fail to observe one or more of the maxims Grice listed,” writes Stanford mathematician Keith Devlin. “The maxims seem more a matter of an obligation of some kind.” In Grice’s own words, “I would like to be able to think of the standard type of conversational practice not merely as something which all or most do in fact follow, but as something which it is reasonable for us to follow, which we should not abandon.”

(Keith Devlin, “What Will Count as Mathematics in 2100?”, in Bonnie Gold and Roger A. Simons, eds., Proof & Other Dilemmas: Mathematics and Philosophy, 2008.)