The Right Stuff

Take any two rational numbers whose product is 2, and add 2 to each. The results are the legs of a right triangle with rational sides.

For example, 13/17 × 34/13 = 2. If we add 2 to each of these we get 47/17 and 60/13 or, clearing fractions, 611 and 1020. The hypotenuse is 1189.

Because (z + 2)2 + [(2/z) + 2]2 = [z + 2 + (2/z)]2, if z is rational, so are all three sides.

(R.S. Williamson, “A Formula for Rational Right-Angle Triangles,” Mathematical Gazette 37:322 [December 1953], 289-290, via Claudi Alsina and Roger B. Nelsen, Icons of Mathematics, 2011.)


I remember, many years ago, when my imagination was warm, and I happened to be in melancholy mood, it distressed me to think of going into a state of being in which Shakespeare’s poetry did not exist. A lady whom I then much admired, a very amiable woman, humored my fancy, and relieved me by saying, ‘The first thing you will meet in the other world will be an elegant copy of Shakespeare’s works presented to you.’ Dr. Johnson smiled benignantly at this, and did not appear to disapprove of the notion.

— James Boswell, Life of Samuel Johnson, 1791

A Trojan Cow

On the evening of July 7, 1969, guards at the Berlin Wall stopped a van headed out of East Berlin. It was carrying a life-size cow that the workmen said would be used in a display in West Berlin. Inside the cow the guards found 18-year-old Angelika B. of Karl-Marx-Stadt. Her fiancé in West Berlin had paid her two accomplices 5,000 DM to smuggle her out; if they’d been successful they’d have received twice that. All three were arrested, and the two helpers were sentenced to three years in prison for “subversive people trafficking.” Angelika was sentenced to 2 years 10 months but was later ransomed by West Germany.

Apparently the cow had been used twice before, successfully, to bring escapees to the West.

Once in a Lifetime

L.E. Dickson on why he’d spent a decade writing a 1,602-page history of number theory: “It fitted with my conviction that every person should aim to perform at some time in his life some serious useful work for which it is highly improbable that there will be any reward whatever other than his satisfaction therefrom.”

“Yes, that man has missed something who has never awakened in an anonymous bed beside a face he will never see again, and who has never left a brothel at sunrise feeling like throwing himself into the river out of pure disgust for life.” — Flaubert

In writing obituaries, “act on the theory that any man has had at least one interesting thing happen to him.” — William S. Maulsby, Getting the News, 1925

The Wooden Horse

British inventor Richard Lovell Edgeworth (1744-1817) could be stunningly imaginative:

I was riding one day in a country, that was enclosed by walls of an uncommon height; and upon its being asserted, that it would be impossible for a person to leap such walls, I offered for a wager to produce a wooden horse, that should carry me safely over the highest wall in the country. It struck me, that, if a machine were made with eight legs, four only of which should stand upon the ground at one time; if the remaining four were raised up into the body of the machine, and if this body were divided into two parts, sliding, or rather rolling on cylinders, one of the parts, and the legs belonging to it, might in two efforts be projected over the wall by a person in the machine; and the legs belonging to this part might be let down to the ground, and then the other half of the machine might have its legs drawn up, and be projected over the wall, and so on alternately. This idea by degrees developed itself in my mind, so as to make me perceive, that as one half of the machine was always a road for the other half, and that such a machine never rolled upon the ground, a carriage might be made, which should carry a road for itself. It is already certain, that a carriage moving on an iron rail-way may be drawn with a fourth part of the force requisite to draw it on a common road.

This seems to anticipate the caterpillar track, and the tank, in the 1760s. He worked on this idea for 40 years, making more than 100 working models and even patenting the principle. Finally he let the patent expire, as it just wasn’t possible with the technology that was available to him. But “I am still satisfied that it is feasible. The experience, which I have acquired by this industry, has overpaid me for the trifling disappointments I have met with; and I have gained far more in amusement, than I have lost by unsuccessful labor.”

(From his memoirs.)

In a Word

n. respects or compliments

adj. appropriate; suitable; proper; fit

n. thorough care or attention

adj. royal

At the Athens Olympics of 1896, American runner Thomas Curtis asked his French competitor Albin Lermusiaux why he was putting on white gloves before the start of the 100-meter race.

Lermusiaux said, “Because I am running in front of the king.”

Containing an Arc

arc puzzle

University of Illinois mathematician John Wetzel called this one of his favorite problems in geometry. Call a plane arc special if it has length 1 and lies on one side of the line through its end points. Prove that any special arc can be contained in an isosceles right triangle of hypotenuse 1.

Click for Answer

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The Map of Discernment

In To Predict Is Not to Explain, mathematician René Thom describes a lunch at which psychoanalyst Jacques Lacan responded strongly to his statement that “Truth is not limited by falsity, but by insignificance.” Thom described the idea later in a drawing:

At the base, one finds an ocean, the Sea of the Insignificance. On the continent, Truth is on one side, Falsehood on the other. They are separated by a river, the River of Discernment. It is indeed the faculty of discernment that separates truth from falsehood. It’s Aristotle’s notion: the capacity for contradiction. It’s what separates us from animals: When information is received by them, it’s instantly accepted and it triggers obedience to its message. Human beings, however, have the capacity to withdraw and to question its veracity.

Following the banks of this river, which flows into the Sea of Insignificance, one travels along a coastline that is slightly concave: Situated at one end is the Slough of Ambiguity; at the other end is the Swamp of La Palice. At the head of the river delta, one sees the Stronghold of Tautology: That’s the stronghold of the logicians. One climbs a rampart towards a small temple, a kind of Parthenon: that’s Mathematics.

To the right, one finds the Exact Sciences: Up in the mountains that surround the bay is Astronomy, with an observatory topping its temple; at the far right stand the giant machines of Physics, the accelerator rings at CERN; the animals in their cages indicate the laboratories of Biology. Out of all this, there emerges a creek that feeds into the Torrent of Experimental Science, which flows into the Sea of Insignificance.

To the left is a wide path climbing towards the north west, up to the City of Human Arts and Sciences. Continuing along it one comes to the foothills of Myth. We’ve entered the kingdom of anthropology. Up at the top is the High Plateau of the Absurd. The spine signifies the loss of the ability to discern contraries, something like an excess of universal understanding which makes life impossible.

He explains the central idea in more detail starting on page 173 of the PDF linked above. “It’s something I’ve done to amuse myself, but it reflects something real, I think: The Logos, the possibility of representation by language, only comes into play for humanity in a rather limited number of situations … [O]ne begins to manufacture linguistic entities which do not correspond to real things. … That’s where the River of Discernment runs into the Fortress of Tautology, into the sewers. It’s become invisible, but at the surface it can smell pretty bad.”