## Curves of Constant Width

Trap a circle inside a square and it can turn happily in its prison — a circle has the same breadth in any orientation.

Perhaps surprisingly, circles are not the only shapes with this property. The Reuleaux triangle has the same width in any orientation, so it can perform the same trick:

In fact any square can accommodate a whole range of “curves of constant width,” all of which have the same perimeter (π*d*, like the circle). Some of these are surprisingly familiar: The heptagonal British 20p and 50p coins and the 11-sided Canadian dollar coin have constant widths so that vending machines can recognize them. What other applications are possible? In the June 2014 issue of the *Mathematical Intelligencer*, Monash University mathematician Burkard Polster notes that a curve of constant width can produce a bit that drills square holes:

… and a unicycle with bewitching wheels:

The self-accommodating nature of such shapes permits them to take part in fascinating “dances,” such as this one among seven triangles:

This inspired Kenichi Miura to propose a water wheel whose buckets are Reuleaux triangles. As the wheel turns, each pair of adjacent buckets touch at a single point, so that no water is lost:

Here’s an immediately practical application: Retired Chinese military officer Guan Baihua has designed a bicycle with non-circular wheels of constant width — the rider’s weight rests on top of the wheels and the suspension accommodates the shifting axles:

(Burkard Polster, “Kenichi Miura’s Water Wheel, or the Dance of the Shapes of Constant Width,” *Mathematical Intelligencer*, June 2014.)

## Holdouts

Only three countries have not officially adopted the metric system: Liberia, Myanmar, and the United States.

In October 2013 Myanmar announced that it plans to make the switch.

## Chinese Magic Mirrors

During China’s Han dynasty, artisans began casting solid bronze mirrors with a perplexing property. The front of each mirror was a polished, reflective surface, and the back featured a design that had been cast into the bronze. But if light were cast from the mirrored side onto a wall, the design would appear there as if by magic.

The mirrors first came to the attention of the West in the early 19th century, and their secret eluded investigators for 100 years until British physicist William Bragg worked it out in 1932. Each mirror had been cast flat with the design on the reverse side, giving the disk a varying thickness. As the front was polished to produce a convex mirror, the thinner parts of the disk bulged outward slightly. These imperfections are invisible to direct inspection; as Bragg wrote, “Only the magnifying effect of reflection makes them plain.”

Joseph Needham, the historian of ancient Chinese science, calls this “the first step on the road to knowledge about the minute structure of metal surfaces.”

## Turing’s Paintbrush

Shortly after joining the faculty of UC San Diego in 1968, British artist Harold Cohen asked, “What are the minimum conditions under which a set of marks functions as an image?” He set out to answer this by writing a computer program that would create original artistic images.

The result, which he dubbed AARON, has been drawing new images since 1973, first still lifes, then people, then full interior scenes with color. These have been exhibited in galleries throughout the world.

Carnegie Mellon philosopher David E. Carrier writes, “A majority of the viewers of AARON’s work find recognizable shapes in it; the drawing above appears to contain human figures. But AARON here used only the twenty or thirty rules it usually uses, with no special reference to human beings. Does knowing this tell us something about the structure of representation?”

Cohen asks, “If what AARON is making is not art, what is it exactly, and in what ways, other than its origin, does it differ from the ‘real thing?’ If it is not thinking, what exactly is it doing?”

“At the risk of stating the obvious, it seems to me that one of the things human beings find interesting about drawings in general is that they are made by other human beings, and here you are watching the image develop *as if* it is being developed by another human being. … When the drawing is finished, it functions as a human drawing. … A large part of what we value in art is not the ability of the artist to communicate special meanings, but rather the ability of the artist to present the viewer with something that stimulates the viewer’s own propensity to generate meaning.”

## A Tidy Theorem

If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (above, *q* = *p* + *r*).

(A corollary of Ptolemy’s theorem.)

## Sad Magic

The magic square at upper left arranges the numbers 3-11 so that each row, column, and long diagonal totals 21.

Lee Sallows found nine tragic words that vary in length from 3 to 11 letters and arranged them into the same square — and he found a unique shape for each word so that every triplet can be assembled into the same 3×7 shape, shown in the border.

## Team Spirit

Thomas Huxley’s *Evolution and Ethics* took China by storm — phrases such as *the strong are victorious and the weak perish* resonated in the national consciousness and “spread like a prairie fire, setting ablaze the hearts and blood of many young people,” noted philosopher Hu Shih.

People even adopted Darwin’s ideas as names. “The once famous General Chen Chiung-ming called himself ‘Ching-tsun’ or ‘Struggling for Existence.’ Two of my schoolmates bore the names ‘Natural Selection Yang’ and ‘Struggle for Existence Sun.’

“Even my own name bears witness to the great vogue of evolutionism in China. I remember distinctly the morning when I asked my second brother to suggest a literary name for me. After only a moment’s reflection, he said, ‘How about the word shih [fitness] in the phrase “Survival of the Fittest”?’ I agreed and, first using it as a nom de plume, finally adopted it in 1910 as my name.”

(Hu Shih, *Living Philosophies*, 1931.)

## The Pythagoras Paradox

Draw a right triangle whose legs *a* and *b* each measure 1. Draw *d* and *e* to complete a unit square. Clearly *d* + *e* = 2.

Now if we cut a “step” into the square as shown, then *f* + *h* = 1 and *g* + *i* = 1, so the total length of the “staircase” is still 2. Cut still finer steps and *j* + *k* + *l* + *m* + *n* + *o* + *p* + *q* is likewise 2.

And so on: The more finely we cut the steps, the more closely their shape approximates that of the original triangle’s diagonal. Yet the total length of the stairstep shape remains 2, the sum of its horizontal and vertical elements. At the limit, then, it would seem that *c* must measure 2 … but we know that the length of a unit square’s diagonal is the square root of 2. Where is the error?

(Thanks, Alex.)

## Round Numbers

A curiosity attributed to a Professor E. Ducci in the 1930s:

Arrange four nonnegative integers in a circle, as above. Now construct further “cyclic quadruples” of integers by subtracting consecutive pairs, always subtracting the smaller number from the larger. So the quadruple above would produce 22, 8, 38, 8, then 14, 30, 30, 14, and so on.

Ducci found that eventually four equal numbers will occur.

A proof appears in Ross Honsberger’s *Ingenuity in Mathematics* (1970).

## Turn, Turn, Turn

The Hoover Dam contains a star map depicting the sky of the Northern Hemisphere as it appeared at the moment that Franklin Roosevelt dedicated the dam. Artist Oskar Hansen imagined that the massive structure might outlive our civilization, and that the map could help future astronomers to calculate the date of its creation. The center star on the map, Alcyone, is the brightest star in the Pleiades, and our sun occupies a position at the center of a flagpole. The whole map traces a complete sidereal revolution of the equinox, a period of 25,694 of our years, and marks the point of the dam’s dedication in that period.

“Man has always sought to express and preserve the magnitude of his exploits in symbols,” Hansen said in 1935. “The written words are symbols arranged so as to preserve in objectified form the thought of man and to record his variant states, both mental and physical. All other arts are similar as to their symbolic significance. They take their place among the category of human endeavor simply as the interpreter of life to itself. They serve as an outer object typifying the inner process. They form the connecting link between the spiritual and the material world. They are the shadows cast by the realities of the soul.”

## Misc

- Juneau, Alaska, is larger than Rhode Island.
- After reading Coleridge’s
*Biographia Literaria*, Byron said, “I wish he would explain his explanation.” - If A + B + C = 180°, then tan A + tan B + tan C = (tan A)(tan B)(tan C).
- Five counties meet in the middle of Lake Okeechobee.
- “Life resembles a novel more often than novels resemble life.” — George Sand

No one knows whether Andrew Jackson was born in North Carolina or South Carolina. The border hadn’t been surveyed well at the time.

## Scoop

When Glenn Seaborg appeared as a guest scientist on the children’s radio show *Quiz Kids* in 1945, one of the children asked whether any new elements, other than plutonium and neptunium, had been discovered at the Metallurgical Laboratory in Chicago during the war.

In fact two had — Seaborg announced for the first time anywhere that two new elements, with atomic numbers 95 and 96 (americium and curium), had been discovered. He said, “So now you’ll have to tell your teachers to change the 92 elements in your schoolbook to 96 elements.”

In his 1979 Priestley Medal address, Seaborg recalled that many students apparently did bring this knowledge to school. And “judging from some of the letters I received from such youngsters, they were not entirely successful in convincing their teachers.”

## Triplets

A pretty new theorem by Lee Sallows: Connect each vertex of a triangle to the midpoint of the opposite side, and place a hinge at that point. Now rotate the smaller triangles about these hinges and you’ll produce three congruent triangles.

If the original triangle is isosceles (or equilateral), then the three resulting triangles will be too.

The theorem appears in the December 2014 issue of *Mathematics Magazine*.

## The Trojan Fly

Achilles overtakes the tortoise and runs on into the sunset, exulting. As he does so, a fly leaves the tortoise’s back, flies to Achilles, then returns to the tortoise, and continues to oscillate between the two as the distance between them grows, changing direction instantaneously each time. Suppose the tortoise travels at 1 mph, Achilles at 5 mph, and the fly at 10 mph. An hour later, where is the fly, and which way is it facing?

Strangely, the fly can be anywhere between the two, facing in either direction. We can find the answer by running the scenario backward, letting the three participants reverse their motions until all three are again abreast. The right answer is the one that returns the fly to the tortoise’s back just as Achilles passes it. But *all* solutions do this: Place the fly anywhere between Achilles and the tortoise, run the race backward, and the fly will arrive satisfactorily on the tortoise’s back at just the right moment.

This is puzzling. The conditions of the problem allow us to predict exactly where Achilles and the tortoise will be after an hour’s running. But the fly’s position admits of an infinite number of solutions. Why?

(From University of Arizona philosopher Wesley Salmon’s *Space, Time, and Motion*, after an idea by A.K. Austin.)

## Traffic Waves

In 2008, physicist Yuki Sugiyama of the University of Nagoya demonstrated why traffic jams sometimes form in the absence of a bottleneck. He spaced 22 drivers around a 230-meter track and asked them to proceed as steadily as possible at 30 kph, each maintaining a safe distance from the car ahead of it. Because the cars were packed quite densely, irregularities began to appear within a couple of laps. When drivers were forced to brake, they would sometimes overcompensate slightly, forcing the drivers behind them to overcompensate as well. A “stop-and-go wave” developed: A car arriving at the back of the jam was forced to slow down, and one reaching the front could accelerate again to normal speed, producing a living wave that crept backward around the track.

Interestingly, Sugiyama found that this phenomenon arises predictably in the real world. Measurements on various motorways in Germany and Japan have shown that free-flowing traffic becomes congested when the density of cars reaches 40 vehicles per mile. Beyond that point, the flow becomes unstable and stop-and-go waves appear. Because it’s founded in human reaction times, this happens regardless of the country or the speed limit. And as long as the total number of cars on the motorway doesn’t change, the wave rolls backward at a predictable 12 mph.

“Understanding things like traffic jams from a physical point of view is a totally new, emerging field of physics,” Sugiyama told Gavin Pretor-Pinney for *The Wavewatcher’s Companion*. “While the phenomenon of a jam is so familiar to us, it is still too difficult to truly understand why it happens.”

## The Facts

“Boarding-House Geometry,” by Stephen Leacock:

Definitions and Axioms

All boarding-houses are the same boarding-house.

Boarders in the same boardinghouse and on the same flat are equal to one another.

A single room is that which has no parts and no magnitude.

The landlady of a boarding-house is a parallelogram — that is, an oblong angular figure, which cannot be described, but which is equal to anything.

A wrangle is the disinclination of two boarders to each other that meet together but are not in the same line.

All the other rooms being taken, a single room is said to be a double room.

Postulates and Propositions

A pie may be produced any number of times.

The landlady can be reduced to her lowest terms by a series of propositions.

A bee line may be made from any boarding-house to any other boarding-house.

The clothes of a boarding-house bed, though produced ever so far both ways, will not meet.

Any two meals at a boarding-house are together less than two square meals.

If from the opposite ends of a boarding-house a line be drawn passing through all the rooms in turn, then the stovepipe which warms the boarders will lie within that line.

On the same bill and on the same side of it there should not be two charges for the same thing.

If there be two boarders on the same flat, and the amount of side of the one be equal to the amount of side of the other, each to each, and the wrangle between one boarder and the landlady be equal to the wrangle between the landlady and the other, then shall the weekly bills of the two boarders be equal also, each to each.

For if not, let one bill be the greater. Then the other bill is less than it might have been — which is absurd.

From his *Literary Lapses*, 1918. See Special Projects.

## A Late Contribution

A ghost co-authored a mathematics paper in 1990. When Pierre Cartier edited a Festschrift in honor of Alexander Grothendieck’s 60th birthday, Robert Thomas contributed an article that was co-signed by his recently deceased friend Thomas Trobaugh. He explained:

The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, ‘The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.’ Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K

_{0}obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.

Thomason himself died suddenly five years later of diabetic shock, at age 43. Perhaps the two are working again together somewhere.

(Robert Thomason and Thomas Trobaugh, “Higher Algebraic K-Theory of Schemes and of Derived Categories,” in P. Cartier et al., eds., *The Grothendieck Festschrift Volume III*, 1990.)

## Cross Purposes

The daily *New York Times* crossword puzzle fills a grid measuring 15×15. The smallest number of clues ever published in a *Times* puzzle is 52 (on Dec. 23, 2008), and the largest is 86 (on Jan. 21, 2005).

This set Bloomsburg University mathematician Kevin Ferland wondering: What are the theoretical limits? What are the shortest and longest clue lists that can inform a standard 15×15 crossword grid, using the standard structure rules (connectivity, symmetry, and 3-letter words minimum)?

The shortest is straightforward: A blank grid with no black squares will be filled with 30 15-letter words, 15 across and 15 down.

The longest is harder to determine, but after working out a nine-page proof Ferland found that the answer is 96: The largest number of clues that a *Times*-style crossword will admit is 96, using a grid such as the one above.

In honor of this result, he composed a puzzle using this grid — it appears in the June-July 2014 issue of the *American Mathematical Monthly*.

(Kevin K. Ferland, “Record Crossword Puzzles,” *American Mathematical Monthly* 121:6 [June-July 2014], 534-536.)

## All for One

A flock of starlings masses near sunset over Gretna Green in Scotland, preparatory to roosting after a day’s foraging. The flock’s shape has a mesmerisingly fluid quality, flowing, stretching, rippling, and merging with itself. Similarly massive flocks form over Rome and over the marshlands of western Denmark, where more than a million migrating starlings form an enormous display known as the “black sun.”

What rules produce this behavior? In the 1970s scientists thought that the birds might be following an electrostatic field produced by the leader. Earler, in the 1930s, one paper even suggested that they use thought transference.

But in 1986 computer graphics expert Craig Reynolds found that he could create a lifelike virtual flock (below) using a surprisingly simple set of rules: direct each bird to avoid crowding nearby flockmates, steer toward the average heading of nearby flockmates, and move toward the center of mass of nearby flockmates.

Studies with real birds seem to bear this out: Under rules like these a flock can react sensitively to a change in direction by any of its members, permitting the whole group to respond efficiently as one organism. “News of a predator’s approach can be communicated rapidly through the flock by whichever of the hundreds of birds on the outside notice it first,” writes Gavin Pretor-Pinney in *The Wavewatcher’s Companion*. “When under attack by a peregrine falcon, for instance, starling flocks will contract into a ball and then peel away in a ribbon to distract and confuse the predator.”

## Versatile

Utica College mathematician Hossein Behforooz devised this “permutation-free” magic square in 2007:

Each row, column, and long diagonal totals 2775, and this remains true if the digits within all 25 cells are permuted in the same way — for example, if we exchange the first two digits of each number, changing 231 to 321, etc., the square retains its magic sum of 2775. Further:

231 + 659 + 973 + 344 + 568 = 2775

979 + 234 + 653 + 341 + 568 = 2775

231 + 343 + 568 + 654 + 979 = 2775

564 + 979 + 233 + 348 + 651 = 2775

231 + 654 + 563 + 978 + 349 = 2775

231 + 348 + 654 + 979 + 563 = 2775

And these combinations of cells maintain their magic totals when their contents are permuted in the same way.

(Hossein Behforooz, “Mirror Magic Squares From Latin Squares,” *Mathematical Gazette*, July 2007.)

## Risk Assessment

Between 1868 and 1870, Mark Twain traveled more than 40,000 miles by rail, dutifully buying accident insurance all the while, and never had a mishap. Each morning he bought an insurance ticket, thinking that fate must soon catch up with him, and each day he escaped without a scratch. Eventually “my suspicions were aroused,” he wrote, “and I began to hunt around for somebody that had won in this lottery. I found plenty of people who had invested, but not an individual that had ever had an accident or made a cent. I stopped buying accident tickets and went to ciphering. The result was astounding. *The peril lay not in traveling, but in staying at home.*”

He calculated that American railways moved more than 2 million people each day, sustaining 650 million journeys per year, but that only 1 million Americans died each year of all causes: “Out of this million ten or twelve thousand are stabbed, shot, drowned, hanged, poisoned, or meet a similarly violent death in some other popular way, such as perishing by kerosene lamp and hoop-skirt conflagrations, getting buried in coal mines, falling off housetops, breaking through church or lecture-room floors, taking patent medicines, or committing suicide in other forms. The Erie railroad kills from 23 to 46; the other 845 railroads kill an average of one-third of a man each; and the rest of that million, amounting in the aggregate to the appalling figure of nine hundred and eighty-seven thousand six hundred and thirty-one corpses, die naturally in their beds!”

The answer, then, is to avoid beds. “My advice to all people is, Don’t stay at home any more than you can help; but when you have got to stay at home a while, buy a package of those insurance tickets and sit up nights. You cannot be too cautious.”

(Mark Twain, “The Danger of Lying in Bed,” *The Galaxy*, February 1871.)

*e*-mergence

1!, 22!, 23!, and 24! contain 1, 22, 23, and 24 digits, respectively.

266!, 267!, and 268! contain 2 × 266, 2 × 267, and 2 × 268 digits, respectively.

2,712! and 2,713! contain 3 × 2,712 and 3 × 2,713 digits, respectively.

27,175! and 27,176! contain 4 × 27,175 and 4 × 27,176 digits, respectively.

271,819!, 271,820!, and 271,821! contain 5 × 271,819, 5 × 271,820, and 5 × 271,821 digits, respectively.

2,718,272! and 2,718,273! contain 6 × 2,718,272, and 6 × 2,718,273 digits, respectively.

27,182,807! and 27,182,808! contain 7 × 27,182,807, and 7 × 27,182,808 digits, respectively.

271,828,170! 271,828,171!, and 271,828,172! contain 8 × 271,828,170, 8 × 271,828,171, and 8 × 271,828,172 digits, respectively.

2,718,281,815! and 2,718,281,816! contain 9 × 2,718,281,815, and 9 × 2,718,281,816 digits, respectively.

27,182,818,270! and 27,182,818,271! contain 10 × 27,182,818,270 and 10 × 27,182,818,271 digits, respectively.

271,828,182,830! and 271,828,182,831! contain 11 × 271,828,182,830, and 11 × 271,828,182,831 digits, respectively.

The pattern continues at least this far:

271,828,182,845,904,523,536,028,747,135,266,249,775,724,655!, 271,828,182,845,904,523,536,028,747,135,266,249,775,724,656!, and 271,828,182,845,904,523,536,028,747,135,266,249,775,724,657! contain 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,655, 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,656, and 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,657 digits, respectively.

(By Robert G. Wilson. More at the Online Encyclopedia of Integer Sequences. Thanks, David.)

## Misc

- Seattle is closer to Finland than to England.
- Is a candle flame alive?
- ABANDON is an anagram of A AND NO B.
- tan
^{-1}(1) + tan^{-1}(2) + tan^{-1}(3) = π - “A thing is a hole in a thing it is not.” — Carl Andre

Detractors of Massachusetts governor Endicott Peabody said that three of the state’s towns had been named for him: Peabody, Marblehead, and Athol.

## Fearless

Founded in the 1880s by Manhattan rationalists, the 13 Club held a regular dinner on the 13th of each month, seating 13 members at each table deliberately to laugh at superstition.

“I have given some attention to popular superstitions, and let me tell you that argument is powerless against them,” founding member Daniel Wolff told journalist Philip Hubert in 1890. “They have a grip upon the imagination that nothing but ridicule will lessen.” As an example he cited the tradition that the mirrors must be removed from a room in which a corpse is lying. “Make the experiment yourself, and the next time you are called upon to sit up with a corpse, notice how uncomfortable a mirror will make you feel,” he said. “Of course it is a matter of the imagination, but you can’t reason against it. All the ingrained terrors of six thousand years are in your bones. You walk across the floor and catch a glimpse of yourself in the glass. You start; was there not a spectral something behind you? So you cover it up.”

As honorary members the club recruited 16 U.S. senators, 12 governors, and six Army generals. Robert Green Ingersoll ended one 1886 toast by declaring, “We have had enough mediocrity, enough policy, enough superstition, enough prejudice, enough provincialism, and the time has come for the American citizen to say: ‘Hereafter I will be represented by men who are worthy, not only of the great Republic, but of the Nineteenth Century.'”

But Oscar Wilde, for one, turned them down. “I love superstitions,” he wrote. “They are the colour element of thought and imagination. They are the opponents of common sense. Common sense is the enemy of romance. The aim of your society seems to be dreadful. Leave us some unreality. Don’t make us too offensively sane.”

(Thanks, David.)