Science & Math

Borromean Tribars

borromean tribars

Only the brilliantly inventive Lee Sallows would think of this. The figure above combines Penrose triangles with Borromean rings: Each of the triangles is an impossible object, and they’re united in a perplexing way — although the three are linked together, no two are linked.

(Thanks, Lee.)

Working Afield

sigurya barbulata

Dutch author Leo Lionni devoted most of his career to children’s books, but in 1977 he undertook a weird experiment. Parallel Botany is a catalog of made-up plants, whose made-up features are described by made-up botanists and illustrated by Lionni’s pencil drawings. Sigurya barbulata, at left, is distinguished by its crowning “cephalocarpus”; a specimen discovered in a Mexican pyramid was found to have been metallized into an organic mace, but how this had come about is the subject of “furious debates.”

“The difficulties of applying traditional methods of research to the study of parallel botany stem chiefly from the matterlessness of the plants,” Lionni wrote. “Deprived as they are of any real organs or tissues, their character would be completely indefinable if it were not for the fact that parallel botany is nonetheless botany, and as such it reflects, even if somewhat distantly, many of the most evident features of normal plants.”

Why do all this? Lionni closes with a quote by the made-up Swedish philosopher Erud Kronengaard: “There are two kinds of men, those who are capable of wonder and those who are not. I hope to God that it is the first who will forge our destiny.”

Alison’s Triangle

alison's triangle

I’m not sure who came up with this — this simple diagram reflects all possible true trigonometric identities of the form x ÷ y = z or x × y = z, where x, y, and z are the basic trigonometric functions of the same angle t.

For any three neighboring functions on the perimeter of the star, the product of the ends always equals the middle (e.g., tan t × cos t = sin t) and the middle function divided by one of the end functions is equal to the other end function (e.g., sin t ÷ tan t = cos t and sin t ÷ cos t = tan t). If you memorize the diagram you can reel off a list of 18 simple relations.

I found it in Michael Stueben’s Twenty Years Before the Blackboard, 1998.

Shape Reference

thomas whales

The index to the fourth edition of George Thomas’ Calculus and Analytic Geometry contains an entry for “Whales” on page 188. That page contains no reference to whales, but it does include the figure above.

http://commons.wikimedia.org/wiki/File:Bessel-Hagen,Erich_1920_G%C3%B6ttingen.jpg?uselang=de

German mathematician Erich Bessel-Hagen was often teased for his protruding ears.

In 1923 his colleague Béla Kerékjártó published a book, Vorlesungen Über Topologie, whose index lists a reference to Bessel-Hagen on page 151.

That page makes no mention of Bessel-Hagen, but it does contain this figure:

2011-03-23-shape-reference-2

Is that libel?

Corner Market

corner market diagram

From Martin Gardner, via Michael Stueben: Obtain a slab of gold measuring 10″ x 11″ x 1″. Divide it diagonally and then cut a triangular notch in two corners as shown. Remove these notches as profit, and slide the remaining halves together to produce a new 10″ x 11″ x 1″ slab. The process can be repeated to yield any amount of money you like!

No Attraction

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Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.

If gravity were turned off, would this still be true?

Click for Answer

Shy

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Pretend that you’ve never seen this before and that it’s an actual living person whose personality you’re trying to read. If you look directly at her face, she seems to hesitate, but if you look near it, say beyond her at the landscape, and try to sense her mood, she smiles at you.

In studying this systematically, Harvard neurobiologist Margaret Livingstone found that “if you look at this painting so that your center of gaze falls on the background or her hands, Mona Lisa’s mouth — which is then seen by your peripheral, low-resolution, vision — appears much more cheerful than when you look directly at it, when it is seen by your fine-detail fovea.

“This explains its elusive quality — you literally can’t catch her smile by looking at it. Every time you look directly at her mouth, her smile disappears because your central vision does not perceive coarse image components very well. People don’t realize this because most of us are not aware of how we move our eyes around or that our peripheral vision is able to see some things better than our central vision. Mona Lisa smiles until you look at her mouth, and then her smile fades, like a dim star that disappears when you look directly at it.”

(From her book Vision and Art: The Biology of Seeing, 2002.)

Worth Remembering

William D. Harvey offered this in Omni in 1980 — a mnemonic for spelling mnemonics:

Mnemonics neatly eliminate man’s only nemesis: insufficient cerebral storage.

First Things First

We [Einstein and Ernst Straus] had finished the preparation of a paper and were looking for a paper clip. After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more drawers we came upon a whole box of unused paper clips. Einstein immediately started to shape one of them into a tool to straighten the bent one. When asked what he was doing, he said, ‘Once I am set on a goal, it becomes difficult to deflect me.’

— Ernst Straus, “Memoir,” in A.P. French, ed., Einstein: A Centenary Volume, 1979

(Einstein said to an assistant at Princeton that this was the most characteristic anecdote that could be told of him.)

Unquote

“[John] von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of von Neumann’s advice. It’s made me a very happy man ever since. But it was von Neumann who put the seed in that grew into my active irresponsibility!” — Richard Feynman

He expands on this in Christopher Sykes’ No Ordinary Genius (1994):

“I got the idea of ‘active irresponsibility’ in Los Alamos. We often went on walks, and one day I was with the great mathematician von Neumann and a few other people. I think Bethe and von Neumann were discussing some social problem that Bethe was very worried about. Von Neumann said, ‘I don’t feel any responsibility for all these social problems. Why should I? I’m born into the world, I didn’t make it.’ Something like that. Well, I’ve read von Neumann’s autobiography and it seems to me that he felt perpetually responsible, but at that moment this was a new idea to me, and I caught onto it. Around you all the time there are people telling you what your responsibilities are, and I thought it was kind of brave to be actively irresponsible. ‘Active’ because, like democracy, it takes eternal vigilance to maintain it — in a university you have to perpetually watch out, and be careful that you don’t do anything to help anybody!”

Hans Bethe:

“Feynman somehow was proud of being irresponsible. He concentrated on his science, and on enjoying life. There are some of us — including myself — who felt after the end of the Second World War that we had a great responsibility to explain atomic weapons, and to try and make the government do sensible things about atomic weapons. … Feynman didn’t want to have anything to do with it, and I think quite rightly. I think it would be quite wrong if all scientists worked on discharging their responsibility. You need some number of them, but it should only be a small fraction of the total number of scientists. Among the leading scientists, there should be some who do not feel responsible, and who only do what science is supposed to accomplish.”

Marvin Minsky:

“I must say I have a little of this sense of social irresponsibility, and Feynman was a great inspiration to me — I have done a good deal of it since. There are several reasons for a scientist to be irresponsible, and one of them I take very seriously: people say, ‘Are you sure you should be working on this? Can’t it be used for bad?’ Well, I have a strong feeling that good and bad are things to be thought about by people who understand better than I do the interactions among people, and the causes of suffering. The worst thing I can imagine is for somebody to ask me to decide whether a certain innovation is good or bad.”

Langton’s Ant

http://commons.wikimedia.org/wiki/File:LangtonsAntAnimated.gif

Set an ant down on a grid of squares and ask it to follow two rules:

  1. If you find yourself on a white square, turn 90° right, change the color of the square to black, and move forward one unit.
  2. If you find yourself on a black square, turn 90° left, change the color of the square to white, and move forward one unit.

That’s it. At first the ant will seem to mill around uncertainly, as above, producing an irregular jumble of black and white squares. But after about 10,000 steps it will start to build a “highway,” following a repeating loop of 104 steps that unfolds forever (below). Computer scientist Chris Langton discovered the phenomenon in 1986.

Will this happen even if some of the starting squares are black? So far the answer appears to be yes — in every initial configuration that’s been tested, the ant eventually produces a highway. If there’s an exception, no one has found it yet.

Asked and Answered

A college professor once offered the following creative final exam: Write a suitable final exam for this course and supply a key. The first paper handed in read ‘Final Exam: Write suitable final exam for this course and supply a key. Key: Any reasonable variation of the previous sentence = 100%.’

— Michael Stueben, Twenty Years Before the Blackboard, 1998

Clerihew

Points
Have no parts or joints.
How then can they combine
To form a line?

— J.A. Lindon

Slipping

In 1960 Jane Goodall watched a chimpanzee repeatedly poking pieces of grass into a termite mound in order to “fish” for insects, the first observation of tool use among animals. When she notified anthropologist Louis Leakey of her discovery, he responded with a telegram:

NOW WE MUST REDEFINE TOOL, REDEFINE MAN, OR ACCEPT CHIMPANZEES AS HUMAN.

Night Crossing

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In late March 1938, Antonio Carrelli received a letter and a telegram in short succession. Both were from Ettore Majorana, the brilliant Italian physicist who had recently joined the faculty of the Naples Physics Institute, where Carrelli was director.

The letter read, “Dear Carrelli, I made a decision that has become unavoidable. There isn’t a bit of selfishness in it, but I realize what trouble my sudden disappearance will cause you and the students. For this as well, I beg your forgiveness, but especially for betraying the trust, the sincere friendship and the sympathy you gave me over the past months. I ask you to remind me to all those I learned to know and appreciate in your Institute, especially Sciuti: I will keep a fond memory of them all at least until 11 pm tonight, possibly later too. E. Majorana.”

The telegram had been sent immediately afterward: “Dear Carrelli, I hope you got my telegram and my letter at the same time. The sea rejected me and I’ll be back tomorrow at the Hotel Bologna traveling perhaps with this letter. However, I have the intention of giving up teaching. Don’t think I’m like an Ibsen heroine, because the case is different. I’m at your disposal for further details. E. Majorana.”

On investigation it was found that Majorana had withdrawn all the money from his bank account and taken the night boat from Naples to Palermo on March 23. He had sent both messages from Palermo and then boarded the steamer to return to Naples on the night of March 25.

But there the trail ended. On the return journey Majorana had shared a compartment with a local university professor, but beyond this point no trace of him could be found. His family offered a reward of 30,000 lire for his whereabouts, and Enrico Fermi implored Mussolini for aid, citing the “deep brilliance” of Majorana’s physics, which he compared to those of Galileo and Newton. A police search found no body but offered no clues.

What happened to him? Theories abound: The most natural explanation, that he committed suicide, is discounted by both his family and the bishop of Trapani, citing his strong Catholic faith. (Also, it doesn’t explain the withdrawal of the money.) Other theories contend that he was murdered, that he fled physics because he foresaw the advent of nuclear weapons, that he had a spiritual crisis and joined a monastery, that he became a beggar, and that he moved to South America. No one knows.

(Barry R. Holstein, “The Mysterious Disappearance of Ettore Majorana,” from the Carolina International Symposium on Neutrino Physics, 2008.)

“Opus 34″

A magic square by Lee Sallows. The 16 pieces progress in area from 1 to 16, and those in each row, column, and long diagonal can be assembled to form the same target shape with area 34.

The Road Coloring Problem

http://commons.wikimedia.org/wiki/File:Road_coloring_conjecture.svg

Every road in this little town is a one-way street, and each street is colored either red or blue. This has a helpful effect: If you start at any house in town and follow the sequence blue-red-red three times in a row, you’ll always arrive at the yellow house.

If you follow blue-blue-red three times, you’ll always arrive at the green one.

In 1970 Roy Adler and Benjamin Weiss asked whether it’s always possible to create such a coloring in a given network; in 2009 Avraham Trahtman proved that, within certain constraints, it is.

Pi Without Circles

The sum of the squares of the reciprocals of the positive integers is π2/6.

The sum of their fourth powers is π4/90.

The sum of their sixth powers is π6/945.

The area of the region under the Gaussian curve y = e-x2 is the square root of π.

The probability that two integers chosen at random will have no prime factor in common is 6/π2.

The integer 8 can be written as the sum of two squares of integers, m2 + n2, in four ways, when (m, n) is (2, 2), (2, -2), (-2, 2), or (-2, -2). The integer 7 can’t be written at all as the sum of such squares. Over a very large collection of integers from 1 to n, the average number of ways an integer can be written as the sum of two squares approaches π. Why?

In a Word

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bibliotaph
n. a hoarder of books

In the rare book collection of the archives at Caltech is a copy of Adrien-Marie Legendre’s 1808 text on number theory. It comes from the collection of Eric Temple Bell, who taught mathematics at Caltech from 1926 to 1953. Inside the book is an inscription in Bell’s handwriting:

This book survived the San Francisco Earthquake and Fire of 18 April, 1906. It was buried with about 600 others, in a vacant lot, before the fire reached the spot. The house next door to the lot fell upon the cache; the tar from the roof baked the 4 feet of dirt, covering the books, to brick, and incinerated all but 4 books, of which this is one. Signed: E. T. Bell. Book buried just below Grace Church, at California and Stockton Streets. House number 729 California Street.

During the Great Fire of London in 1666, Samuel Pepys came upon Sir William Batten burying his wine in a pit in his garden. Pepys “took the opportunity of laying all the papers of my office that I could not otherwise dispose of” and later buried “my Parmazan cheese, as well as my wine and some other things.” I don’t know whether he ever recovered them.

Moments of Inspiration

James Watt perfects the steam engine, 1765:

I had gone to take a walk on a fine Sunday afternoon. I had entered the Green and had passed the old washing house. I was thinking up on the engine at the time and had got as far as the herd’s house, when the idea came into my mind that as steam was an elastic body it would rush into a vacuum, and that if a communication were made between the cylinder and an exhausted vessel it would rush into it and might there be condensed without cooling the cylinder. I had not walked farther than the golf house when the whole thing was arranged clearly in my mind.

Charles Darwin realizes why species diverge, 1840s:

I can remember the very spot in the road, whilst in my carriage, when to my joy the solution occurred to me; and this was long after I had come to Down. The solution, as I believe, is that the modified offspring of all dominant and increasing forms tend to become adapted to many and highly diversified places in the economy of nature.

Walter Cannon recognizes the fight-or-flight response, 1911:

As a matter of routine I have long trusted unconscious processes to serve me. … [One] example I may cite was the interpretation of the significance of bodily changes which occur in great emotional excitement, such as fear and rage. These changes — the more rapid pulse, the deeper breathing, the increase in sugar in the blood, the secretion from the adrenal glands — were very diverse and seemed unrelated. Then, one wakeful night, after a considerable collection of these changes had been disclosed, the idea flashed through my mind that they could be nicely integrated if conceived as bodily preparations for supreme effort in flight or in fighting.

William Rowan Hamilton conceives the fundamental formula for quaternions, 1843:

But on the 16th day of the same month — which happened to be a Monday, and a Council day of the Royal Irish Academy — I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery.

Hamilton adds: “Nor could I resist the impulse — unphilosophical as it may have been — to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely,

i2 = j2 = k2 = ijk = -1

which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.” The bridge now bears a permanent plaque marking Hamilton’s achievement (below), and mathematicians undertake an annual walk from Dunsink Observatory to commemorate it.

On the Couch

In the Minnesota Multiphasic Personality Inventory, the subject is asked whether he agrees with a series of statements, such as “There seems to be a lump in my throat much of the time,” “I am not afraid to handle money,” “I feel uneasy indoors,” and “My sleep is fitful and disturbed.” His responses give insights into his personality and psychopathology.

In 2006, poet Katie Degentesh fed these statements into Google and combined the results into a series of poems, which she published as The Anger Scale. Here’s an excerpt from “As a youngster I was suspended from school one or more times for cutting up”:

Everyone knows about Dallas
and its acts of terrifying gorgeousness

a chef in a tall hat piping meringue
discussing the “brain drain”

dropped a slab of concrete on his left foot
before being lured to the guitar

doesn’t recall details of cutting up friend
to create fake masterpiece

Poets Craig Dworkin and Kenneth Goldsmith call this “a ‘pataphysical nosography, evaluating and diagnosing the mental stability of the Internet itself.” But how do we evaluate the results?

Archimedes’ Twin Circles

Pick any three points on a line and use each pair of them to define a semicircle, as shown.

Now draw a perpendicular between the two smaller semicircles.

Circles c1 and c2 will always have the same area.

Community Spirit

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Considerable amusement was excited, a few years ago, by the announcement that a society for mutual autopsy had been formed among the savants of Paris, with a view to advancing knowledge of the structure and physiology of the brain by a correlation of intellectual characteristics with post mortem appearances. The whole thing was generally regarded as a scientific joke of more than ordinary magnitude. But the society appears to have been a genuine fact, and one of its members, M. Asseline, having recently deceased, his brain was carefully examined by his surviving associates, who made a full report of the result to the Anthropological Society of Paris. The following account of the matter is found in Nature, Aug. 14, 1879, p. 377:

‘M. Asseline died in 1878, at the age of 49. He was a republican and a materialist; was possessed of enormous capacity for work, great faculty of mental assimilation, and an extraordinarily retentive memory; and had a gentle, benevolent disposition, keen susceptibilities, refined taste and subtle wit. As a writer he had always displayed great learning, unusual force of style and elegance of diction, and in his intercourse with others he had been unassuming, sensitive and even timid. Yet the autopsy showed such coarseness and thickness of the convolutions that M. Broca pronounced them to be characteristic of an inferior brain. The fossa or depressions, regarded by Gratiolet as a simian character, and as a sign of cerebral inferiority which are often found in women, and in some men of undoubted intellectual inferiority, were very much marked, especially on the left parietooccipital. But the cranial bones were at some points so thin as to be translucent; the cerebral depressions were deeply marked, the frontal suture was not wholly ossified, a decided degree of asymmetry was manifested in the greater prominence of the right frontal, while, moreover, the brain weighed 1,468 grams, i.e., about 60 grains above the average given by M. Broca for M. Asseline’s age. The apparent contradictions between the weight of the brain and the marked character of the parieto-occipital depressions, attracted much attention, and the members of the Société d’Anthropologie have been earnestly invited by M. Hovelacque, in furtherance of science, to join the Société d’Autopsie, to which anthropology is already indebted for many highly important observations. This society is forming a collection of photographs of its members, which are taken in accordance with certain fixed rules.’

Chicago Medical Journal and Examiner, quoted in New Orleans Medical and Surgical Journal, January 1880

A Knotty Problem

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In a 2002 article in Nature, Australian mathematician Burkard Polster concluded that most of us are doing a pretty good job lacing our shoes: “No matter whether you prefer to lace them straight or criss-crossed, you come close to maximizing the total horizontal tension when you pull on the two ends of one of your shoelaces.”

When it comes to tying them, though, we don’t do so well. “A very large number of people, possibly even the majority, do tie their shoe laces much worse than the rest,” Polster wrote in his 2006 book-length followup, The Shoelace Book. Most of us tie a shoe by placing one half-granny knot on top of another, but this can produce either a very unstable granny knot (left) or a very stable reef knot (right), depending on whether the two half-knots have the same or opposite orientation. (It’s not essential that the second half-granny is typically tied with loops; these are omitted in the diagrams.)

“Hundreds of years of trial and error have led to the strongest way of lacing our shoes,” Polster wrote in Nature, “but unfortunately the same cannot be said about the way in which most of us tie our shoelaces — with a granny knot.”

(Burkard Polster, “What is the best way to lace your shoes?” Nature 2002: 476.)