Jump into the sea and look up. The surface above you is dark except for a bright circle that follows you around like a portable skylight. This is Snell’s window: Because light is refracted as it enters the water, the 180-degree world above you is compressed into a tight 97 degrees.
Physicist Robert W. Wood was thinking of this effect when he created a new wide-angle lens in 1906. Fittingly, he called it the fisheye.
In 1990, Spanish philosopher Jon Perez Laraudogoitia submitted an article to Mind entitled “This Article Should Not Be Rejected by Mind.” In it, he argued:
- If statement 1 in this argument is trivially true, then this article should be accepted.
- If statement 1 were false, then its antecedent (“statement 1 in this argument is trivially true”) would be true, which means that statement 1 itself would be true, a contradiction. So statement 1 must be true.
- But that seems wrong, since Mind is a serious journal and shouldn’t publish trivial truths.
- That means statement 1 must be either false or a non-trivial truth. We know it can’t be false (#2), so it must be a non-trivial truth, and its antecedent (“statement 1 in this argument is trivially true”) is false.
- What then is the truth value of its consequent, “this article should be accepted”? If this were false then Mind shouldn’t publish the article; that can’t be right, since the article consists of a non-trivial truth and its justification.
- So the consequent must be true, and Mind should publish the article.
They published it. “This is, I believe, the first article in the whole history of philosophy the content of which is concerned exclusively with its own self, or, in other words, which is totally self-referential,” Laraudogoitia wrote. “The reason why it is published is because in it there is a proof that it should not be rejected and that is all.”
In 1964 Canadian writer Graeme Gibson bought a parrot in Mexico. The bird, which Gibson named Harold Wilson, was bright and affectionate at first, but he seemed to grow lonely in the dark Canadian winter, so in the spring Gibson made arrangements to donate him to the Toronto Zoo. At the aviary Gibson carried Harold into the cage that had been prepared for him, placed him on a perch, said his goodbyes, and turned to go.
“Then Harold did something that astonished me. For the very first time, and in exactly the voice my kids might have used, he called out ‘Daddy!’ When I turned to look at him he was leaning towards me expectantly. ‘Daddy’, he repeated.
“I don’t remember what I said to him. Something about him being happier there, that he’d soon make friends. The kind of things you say to kids when you abandon them at camp. But outside the aviary I could still hear him calling ‘Daddy! Daddy!’ as we walked away. I was shattered to discover that Harold knew my name, and that he did so because he’d identified himself with my children.
“I now believe he’d known it all along, but was using it — for the first time — out of desperation. Both Konrad Lorenz and Bernd Heinrich mention instances of birds calling out the private names of intimates when threatened by serious danger. I am no longer surprised by such information. We think of our captive birds as our pets, but perhaps we are theirs as well.”
(From Gibson’s Perpetual Motion, 1982.)
Inscribe a hexagon in a unit circle such that AB = CD = EF = 1.
Now the midpoints of BC, DE, and FA form an equilateral triangle.
See A Better Nature.
Arthur W.J.G. Ord-Hume calls this “the most graceful and simple perpetual motion machine of all time.” It was offered by American inventor F.G. Woodward in the 19th century. A heavy wheel is mounted between two rollers so that the wheel’s weight causes it to roll continuously in the direction of the arrow.
Or so Woodward hoped. Ord-Hume notes that the principle required the left half of the wheel always to be heavier than the right half. “Sadly, it wasn’t.”
How do I know that I’m not just a fictional character in some imagined story? What could I learn about myself that would prove that I’m real? “I am human, male, brunette, etc., but none of that helps,” writes UCLA philosopher Terence Parsons. “I see people, talk to them, etc., but so did Sherlock Holmes.”
Descartes would say that the very fact that I’m thinking about this shows that I exist: cogito ergo sum. But a fictional character could make the same argument. “Hamlet did think a great many things,” writes Jaakko Hintikka. “Does it follow that he existed?” Robert Nozick adds, “Could not any proof be written into a work of fiction and be presented by one of the characters, perhaps one named ‘Descartes’?”
Tweedledee tells Alice that she’s only a figment of the Red King’s dream. “If that there King was to wake,” adds Tweedledum, “you’d go out — bang! — just like a candle!”
Alice says, “Hush! You’ll be waking him, I’m afraid, if you make so much noise.”
“Well, it’s no use YOUR talking about waking him,” replies Tweedledum, “when you’re only one of the things in his dream. You know very well you’re not real.”
“It seems to me that this is a philosophical problem that deserves to be treated seriously on a par with issues like the reality of the external world and the existence of other minds,” Parsons writes. “I don’t know how to solve it.”
(Terence Parsons, Nonexistent Objects, 1980; Charles Crittenden, Unreality, 1991; Robert Nozick, “Fiction,” Ploughshares 6:3 (1980), pp. 74-78; Jaakko Hintikka, “Cogito, Ergo Sum: Inference or Performance?”, The Philosophical Review, 71:1 (January 1962), pp. 3-32.)
If a flock of birds disperses gradually, at what point does it cease to be a flock?
“There is at the moment a pipe on my desk,” wrote MIT philosopher Richard Cartwright in 1987. “Its stem has been removed, but it remains a pipe for all that; otherwise no pipe could survive a thorough cleaning.”
But he also owned a two-volume set of John McTaggart’s The Nature of Existence, one volume of which was in Cambridge and the other in Boston. Do those two volumes still make one thing? If so, is there a “thing” composed of the Eiffel Tower and the Old North Church? Why not?
(From Cartwright’s Philosophical Essays.)
Draw a circle whose circumference is the golden mean. Choose a point and label it 1, then move clockwise around the circle in steps of arc length 1, labeling the points 2, 3, and so on. At each step, the difference between each pair of adjacent numbers on the circle is a Fibonacci number.
- What time is it at the North Pole?
- The shortest three-syllable word in English is W.
- After the revolution, the French frigate Carmagnole used a guillotine as its figurehead.
- 823502 + 381252 = 8235038125
- PRICES: CRIPES!
- “Conceal a flaw, and the world will imagine the worst.” — Martial
When Montenegro declared independence from Yugoslavia, its top-level domain changed from .yu to .me.
If I buy two toothbrushes in a “buy one, get one free” offer … which one did I buy, and which was free?
(From philosopher Roy Sorensen.)
How long is a coastline? If we measure with a long yardstick, we get one answer, but as we shorten the scale the total length goes up. For certain mathematical shapes, indeed, it goes up without limit.
English mathematician Lewis Fry Richardson discovered this perplexing result in the early 20th century while examining the relationship between the lengths of national boundaries and the likelihood of war. If the Spanish claim that the length of their border with Portugal is 987 km, and the Portuguese say it’s 1,214 km, who’s right? The ambiguity arises because a wiggly boundary occupies a fractional dimension — it’s something between a line and a surface.
“At one extreme, D = 1.00 for a frontier that looks straight on the map,” Richardson wrote. “For the other extreme, the west coast of Britain was selected because it looks like one of the most irregular in the world; it was found to give D = 1.25.”
This is a mathematical notion, but it’s also a practical problem. On the fjord-addled panhandle of Alaska, the boundary with British Columbia was originally defined as “formed by a line parallel to the winding of the coast.” Who gets to define that? On the map below, the United States claimed the blue border, Canada wanted the red one, and British Columbia claimed the green. The yellow border was arbitrated in 1903.
Write out the positive powers of 10 in both base 2 and base 5:
Now for any integer n > 1, we’ll find exactly one number of length n somewhere on the two lists. They contain one 3-digit number, one 4-digit number, and so on forever — if n = 100 we find a 100-digit number in the 30th position on the base 2 list.
(This result first appeared in the 1994 Asian Pacific Mathematics Olympiad. I found it in Ravi Vakil’s A Mathematical Mosaic.)
Two further curious lists: If we write out the triangular numbers, those in positions 3, 33, etc. show a pattern:
T(3) = 6
T(33) = 561
T(333) = 55611
T(3,333) = 5556111
T(33,333) = 555561111
T(333,333) = 55555611111
T(6) = 21
T(66) = 2211
T(666) = 222111
T(6,666) = 22221111
T(66,666) = 2222211111
T(666,666) = 222222111111
Draw any triangle and divide each leg into three equal segments. Connect each vertex to one of the trisection points on the opposite leg, as shown, and the triangle formed in the center will have 1/7 the area of the original triangle.
A square inscribed in a semicircle has 2/5 the area of a square inscribed in a circle of the same radius.
Draw a square and connect each vertex to the midpoint of an opposite side, as shown. The square formed in the center will have 1/5 the area of the original square.
A “proof without words”:
Steven Bartlett and Peter Suber’s Self-Reference: Reflections on Reflexivity contains a bibliography of works on reflexivity.
It includes an entry for Steven Bartlett and Peter Suber’s Self-Reference: Reflections on Reflexivity.
Until 1911, the U.S. House of Representatives grew along with the country. Accordingly, when the 1880 census showed an increase in population, C.W. Seaton, chief clerk of the census office, worked out apportionments for all House sizes between 275 and 350, in order to see which states would get the new seats.
He was in for a surprise. The method was straightforward: Take the total U.S. population and divide it by the proposed number of seats in the House, rounding all fractions down. This would dispose of most of the seats; any leftover seats would be awarded to the states whose fractional remainders had been highest. But Seaton discovered an oddity:
If the House had 299 seats, Alabama would get 8 representatives (because its remainder, .646, was higher than that of Texas or Illinois). But if the House had 300 seats it would get only 7 (the extra representative would now go to Illinois, whose remainder had surpassed Alabama’s). The problem is that the “fair share” of a large state increases more quickly than that of a small state.
Seaton called this the Alabama paradox. A related problem is the population paradox: If the method above had been used in 1901 to reallocate 386 seats in the House, Virginia would have lost a seat to Maine even though the ratio of their populations had increased from 2.67 to 2.68:
Here, even though the size of the House has not changed, a fast-growing state receives fewer representatives than a slow-growing one.
In 1982 mathematicians Michel Balinski and Peyton Young showed that if each party gets one of the two numbers closest to its fair share of seats, then any system of apportionment will run into one of these paradoxes. The solution, it seems clear, is to start cutting legislators into pieces.
(These data are from Hannu Nurmi’s Voting Paradoxes and How to Deal With Them, 1999. Balinski and Young’s book is Fair Representation: Meeting the Ideal of One Man, One Vote.)
In 1981, when science journalist Marcus Chown was an undergraduate physics student, his mother watched a profile of Richard Feynman on the BBC series Horizon. She had never shown an interest in science before, and he wanted to encourage her, so when he advanced to Caltech to study astrophysics, he told Feynman of his mother’s interest and asked him to send her a birthday note. She received this:
Happy Birthday Mrs. Chown!
Tell your son to stop trying to fill your head with science — for to fill your heart with love is enough!
Richard P. Feynman (the man you watched on BBC “Horizons”)
Male bees come from unfertilized eggs, so they have mothers but no fathers. Females come from fertilized eggs, so they have parents of both sexes. This produces an interesting pattern: The number of males in a given generation equals the number of females in the succeeding generation. And the number of females in a given generation equals the number of females in the succeeding two generations:
So the total number of bees, male and female, in generation n is the Fibonacci number Fn.
W. Hope-Jones discovered the relationship in 1921; this example is from Thomas Koshy’s Fibonacci and Lucas Numbers With Applications, 2001.
The first 10 digits of the golden ratio φ can be rearranged to give the first 10 digits of 1/π:
φ = 1.618033988 …
1/π = .3183098861 …
And the first nine digits of 1/φ can be rearranged to give the first 9 digits of 1/π:
1/φ = .618033988 …
1/π = .318309886 …
In 1983 amateur mathematician George Odom discovered that if points A and B are the midpoints of sides EF and DE of an equilateral triangle, and line AB meets the circumscribing circle at C, then AB/BC = AC/AB = φ. Odom used this fact to construct a pentagon, which H.S.M. Coxeter published in the American Mathematical Monthly with the single word “Behold!”
In 2011 Australian architect Horst Kiechle created an entire human torso from paper, as a geometric sculpture, for the science lab at the International School Nadi in Fiji.
He’s made the templates available for free — you can fold your own paper man, complete with removable organs.
Completed in 1997, German artist Jo Niemeyer’s 20 Steps Around the Globe installed 20 high-grade steel columns on a great circle around the earth, establishing the distances between them using the golden ratio φ, 1.61803398875.
The first poles, shown here, were erected in Finnish Lapland, north of the polar circle. The first two were placed 0.458 meters apart; the third was placed 0.458 × φ = 0.741 meters beyond the second; and so on, marching off in a beeline toward the horizon. The first 12 poles are in Finland; the 13th and 14th in Norway; the 15th, 16th, and 17th in Russia; the 18th in China; and the 19th in Australia. The 20th coincides with the first back in Finland.
In this way the project models the golden section and the Fibonacci sequence, tailoring them to our planet. Niemeyer calls it “an interdisciplinary expedition into the secrets of the power of limits.”
In 1680 Robert Hooke sprinkled a plate with flour, drew a violin bow across its edge, and saw the flour spring into surprising geometric shapes. The plate was resonating, driving the flour into invisible nodal lines on its surface that were not vibrating.
German physicist Ernst Chladni pursued these experiments in the 18th century and published his results in Discoveries in the Theory of Sound in 1787. Today they’re known as Chladni figures.
“The universe is full of magical things,” wrote Eden Phillpotts, “patiently waiting for our wits to grow sharper.”
In 1972 Canadian scientists R.W. Sheldon and S.R. Kerr set out to reason out the number of monsters that occupy Loch Ness. Because the creatures are reportedly large and rarely seen, it follows that their numbers must be small. (“It has been suggested from time to time that as the monsters are never caught it must therefore follow that they do not exist. This is both irresponsible and illogical.”)
By estimating the fish stock available in the loch, they determined that the total mass of monsters is between 3,135 and 15,675 kg. Taking the minimum monster size as 100 kg (“anything smaller is not suitably monstrous”), they estimate that the loch contains between 1 and 156 monsters. The high end of this range seems unlikely; and since monsters have been reported for centuries they’re probably breeding, which would require a population of at least 10.
Given the available quantity of fish and assuming a stable population, monsters weighing 100 kg would have to die at a rate of at least 3 per year. Larger animals would die less frequently, and this seems likely since dead monsters are never found (and since the juveniles that must replace them are never seen). So it seems the lake probably contains a small number of large monsters, perhaps 10-20 monsters weighing up to 1,500 kg each and measuring about 8 meters, “a size that agrees well with observational data.”
“We would like to thank Kate Kranck for drawing our attention to this problem, because until she mentioned it we were unaware that monsters were a problem.”
(“The Population Density of Monsters in Loch Ness,” Limnology and Oceanography 17:5, 796–798)
Label the faces of a fair set of dice with these numbers:
Die A: 3, 3, 3, 3, 3, 6
Die B: 2, 2, 2, 5, 5, 5
Die C: 1, 4, 4, 4, 4, 4
This gives them a curious property. In the long run Die A will tend to beat Die B, Die B will tend to beat Die C, and Die C will tend to beat Die A. The three dice form a ring in which each die beats its successor. No matter which die our opponent chooses, we can select another that is likely to beat it.
Business magnate Warren Buffet once challenged Bill Gates to such a game using four nontransitive dice. “Buffett suggested that each of them choose one of the dice, then discard the other two,” wrote Janet Lowe in her 1998 book Bill Gates Speaks. “They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates’s curiosity. He asked to examine the dice.”
“It wasn’t immediately evident that because of the clever selection of numbers for the dice, they were nontransitive,” Gates said. “Assuming the dice were rerolled, each of the four dice could be beaten by one of the others.” He invited Buffett to choose first.
Facilities suggested by Lewis Carroll for a school of mathematics at Oxford, 1868:
- A very large room for calculating Greatest Common Measure. To this a small one might be attached for Least Common Multiple: this, however, might be dispensed with.
- A piece of open ground for keeping Roots and practising their extraction: it would be advisable to keep Square Roots by themselves, as their corners are apt to damage others.
- A room for reducing Fractions to their Lowest Terms. This should be provided with a cellar for keeping the Lowest Terms when found, which might also be available to the general body of Undergraduates, for the purpose of “keeping Terms.”
- A large room, which might be darkened, and fitted up with a magic lantern for the purpose of exhibiting Circulating Decimals in the act of circulation. This might also contain cupboards, fitted with glass-doors, for keeping the various Scales of Notation.
- A narrow strip of ground, railed off and carefully levelled, for investigating the properties of Asymptotes, and testing practically whether Parallel Lines meet or not: for this purpose it should reach, to use the expressive language of Euclid, “ever so far.”
He introduced this topic with an administrator by writing, “Dear Senior Censor,–In a desultory conversation on a point connected with the dinner at our high table, you incidentally remarked to me that lobster-sauce, ‘though a necessary adjunct to turbot, was not entirely wholesome.’ It is entirely unwholesome. I never ask for it without reluctance: I never take a second spoonful without a feeling of apprehension on the subject of possible nightmare. This naturally brings me to the subject of Mathematics …”