Science & Math

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.)

Overthinking

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When I was a child, it was believed that animals became extinct because they were too specialized. My father used to tell us about the saber-tooth tiger’s teeth — how they got too big and the tiger couldn’t eat because he couldn’t take game anymore. And I remember my father saying, with my brother sitting there, ‘I wonder what it will be with the human beings that will be so overspecialized that they’ll kill themselves off?’

My father never found out that my brother was working on the bomb.

— Richard Feynman’s sister Joan, quoted in Christopher Sykes, No Ordinary Genius, 1994

Sweet Home

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I am determined & feel sure, that the scenery of England is ten times more beautiful than any we have seen.– What reasonable person can wish for great ill proportioned mountains, two & three miles high? No, no; give me the Brythen or some such compact little hill.– And then as for your boundless plains & impenetrable forests, who would compare them with the green fields & oak woods of England?– People are pleased to talk of the ever smiling sky of the Tropics: must not this be precious nonsense? Who admires a lady’s face who is always smiling? England is not one of your insipid beauties; she can cry, & frown, & smile, all by turns.– In short I am convinced it is a most ridiculous thing to go round the world, when by staying quietly, the world will go round with you.

— Charles Darwin, letter to his sister, July 18, 1836. He was on board the Beagle, bound for Ascencion. He had written the previous December, “How glad I shall be, when I can say, like that good old Quarter Master, who entering the Channel, on a gloomy November morning, exclaimed, ‘Ah here there are none of those d—-d blue skys’.”

Budget Trouble

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An energetic boy got a piggy bank for his birthday. He decided that from then on he will number every bill he gets from his grandparents (1, 2, …) and put it all in his bank. During the first half year he got 2 bills, but at the end of this period he pulled out 1 bill (chosen at random). In the next 1/4 year he got 2 more bills, but at the end of this period he pulled 1 bill chosen at random from the 3 bills in his bank. In the next 1/8 year he repeated the same routine etc. (each period is half the length of the previous period). What is the probability that any of the bills he got during this year will remain in his bank after a full year of the above activity? Paradoxically the probability is 0, even though it is clear that he only spent half of his money. Can we offer the boy good financial advice without making him cut his expenses?

— Talma Leviatan, “On the Use of Paradoxes in the Teaching of Probability,” Proceedings of ICOTS 6, 2002

The Meaning of Life

Along with art and love, life is one of those bedeviling concepts that we really ought to have a definition for but don’t. Philosophers tend to regard the question as too scientific, and scientists as too philosophical. Linus Pauling observed that it’s easier to study the subject than to define it, and, J.B.S. Haldane noted, “no definition will cover its infinite and self-contradictory variety.”

Classical definitions of life typically refer to structural features, growth, reproduction, metabolism, motion against force, response to stimuli, evolvability, and information content and transfer. But definitions built on these elements are prone to exceptions. Fire grows, moves, metabolizes, reproduces, and responds to stimuli, but is “nonliving.” So are free-market economies and the Internet, which evolve, store representations of themselves, and behave “purposefully.” I am nonreproducing but, I hope, still alive.

If we we look around us, it’s hard to find a property that’s unique to life, and even if we could, our observations are limited to Earth’s biosphere, a tiny, tenuous environment like a film of water on a basketball. But if we expand our list to include abstract properties such as resistance to entropy, then we risk including alien phenomena that we might not regard intuitively as living.

Perhaps the answer is more poetic. “As I see it, the great and distinguishing feature of living things … is that they have needs — continual, and, incidentally, complex needs,” wrote botanist Donald C. Peattie in 1935. “I cannot conceive how even so organized a dead system as a crystal can be said to need anything. But a living creature, even when it sinks into that half-death of hibernation, even the seed in the bottom of the driest Mongolian marsh, awaiting rain through two thousand years, still has needs while there is life in it.”

Extra Credit

cooper malaria watercolors

In studying the parasitic protozoan Plasmodium ovale in 1954, English parasitologist William Cooper volunteered to receive the bites of about a thousand mosquitos, and nine days later underwent a laparotomy in which a piece of his liver was removed. On recovering, he stained the sections himself, located the malaria parasite stages in his own tissue, and painted these in watercolors to accompany the resulting article.

His coauthor, University of London protozoologist Cyril Garnham, wrote that Cooper “attained everlasting fame by this episode.”

(P.C.C. Garnham et al., “The Pre-Erythrocytic Stage of Plasmodium Ovale,” Transactions of the Royal Society of Tropical Medicine and Hygiene 49:2 [March 1955], 158-167)

(Thanks, Andrew.)

Neat

A gram is the mass of one cubic centimeter of water; the earth’s gravitational attraction is approximately 10 in metric units (9.8 meters/second2); and atmospheric pressure works out to about 1 kilogram per square centimeter.

This shows that the pressure under 10 meters of water is about one atmosphere. Ten meters of water is 1000 centimeters, so a column one centimeter square would weigh one kilogram and exert a pressure of 1 kilogram per square centimeter.

(Thanks, Steve.)

Righteousness

The autobiography of the 12th-century Muslim poet Usama ibn Munqidh tells of an incident in which the invading Crusaders appealed for a doctor to treat some of their number who had fallen ill. The Muslims sent a doctor named Thabit, who returned after 10 days with this story:

They took me to see a knight who had an abscess on his leg, and a woman with consumption. I applied a poultice to the leg, and the abscess opened and began to heal. I prescribed a cleansing and refreshing diet for the woman. Then there appeared a Frankish doctor, who said: ‘This man has no idea how to cure these people!’ He turned to the knight and said: ‘Which would you prefer, to live with one leg or die with two?’ When the knight replied that he would prefer living with one leg, he sent for the strong man and a sharp axe. They arrived, and I stood by to watch. The doctor supported the leg on a block of wood, and said to the man: ‘Strike a mighty blow, and cut cleanly!’ … The marrow spurted out of the leg (after the second blow) and the patient died instantaneously. Then the doctor examined the woman and said: ‘She has a devil in her head who is in love with her. Cut her hair off!’ This was done, and she went back to eating her usual Frankish food … which made her illness worse. ‘The devil has got into her brain,’ pronounced the doctor. He took a razor and cut a cross on her head, and removed the brain so that the inside of the skull was laid bare … the woman died instantly. At this juncture I asked whether they had any further need of me, as they had none I came away, having learnt things about medical methods that I never knew before.

Afghan Bands

afghan bands

Most people know that if you cut a Möbius band in two lengthwise you’ll produce one band rather than two. But splitting the ends and joining them also creates some surprising effects.

Starting with the figure above, join ends A and D directly, then pass B under A and join it to E. Now pass C over B and under A; pass F over D and under E; and join C and F. Extend the slits along the length of the band and you’ll have three linked rings.

Now compare this variant, suggested by Ellis Stanyon in 1930: Starting again from the diagram, give E a half-twist to the right and join it to C; give F a half-twist to the right and join it to B; then pass A under B and join it to D (without turning it over). Cut along the two slits and you’ll produce a small ring linked to a large one. What became of the third ring?

There’s a surprisingly simple way to produce a similar effect: Draw a line along the length of a Möbius band, one-third of the way across the strip. Cutting along this line will produce a large band linked to a small one — and this time the small band is itself a Möbius band, on which you can repeat the feat.

Cardiology

In other words, this is the day on which those charming little missives, ycleped Valentines, cross and intercross each other at every street and turning. The weary and all for-spent twopenny postman sinks beneath a load of delicate embarrassments, not his own. It is scarcely credible to what an extent this ephemeral courtship is carried on in this loving town, to the great enrichment of porters, and detriment of knockers and bell-wires. In these little visual interpretations, no emblem is so common as the heart,–that little three-cornered exponent of all our hopes and fears,–the bestuck and bleeding heart; it is twisted and tortured into more allegories and affectations than an opera-hat. What authority we have in history or mythology for placing the head-quarters and metropolis of god Cupid in this anatomical seat rather than in any other, is not very clear; but we have got it, and it will serve as well as any other thing. Else we might easily imagine, upon some other system which might have prevailed for any thing which our pathology knows to the contrary, a lover addressing his mistress, in perfect simplicity of feeling, ‘Madam, my liver and fortune are entirely at your disposal;’ or putting a delicate question, ‘Amanda, have you a midriff to bestow?’ But custom has settled these things, and awarded the seat of sentiment to the aforesaid triangle, while its less fortunate neighbours wait at animal and anatomical distance.

— Charles Lamb, Essays of Elia, 1823

A mathematical valentine:

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Justice Confused

Suppose that a house is robbed and police find a strand of the burglar’s hair at the scene of the crime. A suspect is in custody, and tests show that the strand matches his hair. A forensic scientist testifies that the chance of a random person producing such a match is 1/2000. Does this mean that there’s a 1999/2000 chance that the suspect is guilty?

No, it doesn’t. In a city of 5 million there will be 1/2000 × 5,000,000 = 2,500 people who produce a match, so on the basis of this evidence alone the probability that the suspect is guilty is only 1/2500.

In a 1987 article, William Thomson and Edward Schumann dubbed this “prosecutor’s fallacy.” Unfortunately, it’s matched by the “defense attorney’s fallacy,” which holds that the hair-match evidence is worthless because it increases the likelihood of the suspect’s guilt by a negligible amount, 1/2500. In fact it drastically narrows the range of possible suspects, from 5 million to 2,500, while failing to exclude the defendant, hardly cause for confidence.

Worryingly, Thompson and Schumann found an experienced prosecutor who insisted that if a defendant and a perpetrator match on a blood type found in 10 percent of the population, then there’s a 10 percent chance that the defendant would have this blood type if he were innocent and hence a 90 percent chance that he’s guilty. “If a prosecutor falls victim to this error,” they write, “it is possible that jurors do as well.”

(William C. Thompson and Edward L. Schumann, “Interpretation of Statistical Evidence in Criminal Trials,” Law and Human Behavior, 11:3 [September 1987], 167-187)

Fixing Dates

In 1899, British statistician Moses B. Cotsworth noted that recordkeeping could be greatly simplified if each month contained a uniform number of whole weeks. He proposed an “international fixed calendar” containing 13 months of 28 days each:

international fixed calendar

This makes everything easier. The 26th of every month falls reliably on a Thursday, for example, and statistical comparisons between months are made more accurate, as each month contains four tidy weeks with four weekends. (Unfortunately for the superstitious, every one of the 13 months contains a Friday the 13th.) A new month, called Sol, would be wedged between June and July, and an extra day, “Year Day,” would be added at the end of the year, but it would be independent of any month (as would Leap Day).

In 1922 the League of Nations chose Cotsworth’s plan as the most promising of 130 proposed calendar reforms, but the public, as always, resisted the unfamiliar, and by 1937 the International Fixed Calendar League had closed its doors. It left one curious legacy, though: George Eastman, the founder of Eastman Kodak, was so pleased with Cotsworth’s scheme that he adopted it as his company’s official calendar — and it remained so until 1989.

Misc

  • Babe Ruth struck out 1,330 times.
  • EMBARGO spelled backward is O GRAB ME.
  • The numbers on a roulette wheel add to 666.
  • The fourth root of 2143/22 is nearly pi (3.14159265258).
  • “A prosperous fool is a grievous burden.” — Aeschylus

Six countries have names that begin with the letter K, and each has a different vowel as the second letter: Kazakhstan, Kenya, Kiribati, Kosovo, Kuwait, Kyrgyzstan.

(Thanks, Danny.)

The Ulysses Contract

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In 1982, 24-year-old schizophrenic patient J.S. faced a difficult decision: The neuroleptic drug Prolixin relieved his psychotic symptoms, but it produced tardive dyskinesia, a progressive disorder that caused uncontrollable movements of his legs, arms, and tongue.

His therapist learned of an experimental program that might reduce this side effect, and J.S. signed consent forms to enter treatment. But the first step was to stop all medications, and without the Prolixin he descended again into psychosis and refused the experimental medication.

This produces an impossible dilemma: Does J.S.’ “sane” self have the right to overrule his “insane” self, if the two disagree? Can Dr. Jekyll bind Mr. Hyde? Such a directive is sometimes called a Ulysses contract, after the Greek hero who ordered his men to disregard his commands as they sailed past the sirens. If a patient directs his caregivers to ignore his own future requests, can the caregivers follow these orders?

In J.S.’ case, the answer was no. The research unit’s legal counsel decided that his earlier consent did not override his later refusal, and he was withdrawn from the program. When he resumed his antipsychotic medication and learned what had happened, he begged for another chance to try the experimental medication. Had they been wrong to refuse him?

(Morton E. Winston, Sally M. Winston, Paul S. Appelbaum, and Nancy K. Rhoden, “Can a Subject Consent to a ‘Ulysses Contract’?”, The Hastings Center Report, 12:4 [August 1982], 26-28)

The Six Circles Theorem

Fit a circle into one corner of a triangle. Now fit a second circle into a second corner so that it’s tangent to the first circle. Then fit a third circle into the third corner so that it’s tangent to the second circle.

Keep this up, cycling among the three corners, and the sixth circle will be tangent to the first one.

Dog Tired

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Maybe figures can’t lie, but liars can certainly figure, and that is why statistics can be made to ‘prove’ almost anything. Consider a group of ten girls, nine of them virgins, one pregnant. On the ‘average’ each of the nine virgins is ten per cent pregnant, while the girl who is about to have a baby is ninety per cent a virgin. Or, assuming that a fox terrier two feet long, with a tail an inch and a half high, can dig a hole three feet deep in ten minutes, to dig the Panama Canal in a single year would require only one fox terrier fifteen miles long, with a tail a mile and a half high.

— Stuart Cloete, The Third Way, 1947

Never the Twain

proclus paradox

A paradox attributed to Proclus Lycaeus (412-485):

Consider two nonparallel lines, AQ and BP. BP is perpendicular to AB; AQ isn’t. Find the midpoint of AB and mark AC = BD = AB/2. Now if AQ and BP are going to intersect, it can’t happen on AC or BD; if it did, say at a point R, then that would give us a triangle ARB where the sum AR + RB < AB, which is impossible.

But now we can connect CD and follow the same process: CE and DF can't intersect for the same reason. EG and FH are likewise ruled out, and so on up the line forever.

This seems to mean that two nonparallel lines will never intersect. That can’t be right, but where is the error?

(From Alfred Posamentier, Magnificent Mistakes in Mathematics, 2013.)

Mundane

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Earth is the only planet not named after a god.