Science & Math


Create a strip of 19 triangles like the one above (printable version here) and fold the left portion back successively at each of the northeast-pointing lines to produce a spiral:

Fold this spiral backward along line ab:

Then fold the resulting figure backward at cd. You should be left with one blank triangular tab that can be folded backward and pasted to another blank panel on the opposite side. The resulting hexagon should have six 1s on one side and six 2s on the other.

With some adroit pinching this hexagon produces some marvelous effects. Fold down two adjacent triangles so that they meet, and then press in the opposite corner to join them. Now the top of the figure can be prised open and folded down to produce a new hexagon — this one with 1s on one face and a surprising blank on the second. What has become of the 2s?

Exploring the properties of this “hexahexaflexagon” offers an intuitive lesson in geometric group theory:

When Martin Gardner wrote about these bemusing creatures in his first column for Scientific American in 1956, he received two letters. The first was from Neil Uptegrove of Allen B. Du Mont Laboratories in Clifton, N.J.:


I was quite taken with the article entitled ‘Flexagons’ in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.

But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.

We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of the hexahexaflexagon.

Here is our question: Does his widow draw workmen’s compensation for the duration of his absence, or can we have him declared legally dead immediately? We await your advice.

The second was from Robert M. Hill of The Royal College of Science and Technology in Glasgow, Scotland:


The letter in the March issue of your magazine complaining of the disappearance of a fellow from the Allen B. Du Mont Laboratories ‘down’ a hexahexaflexagon, has solved a mystery for us.

One day, while idly flexing our latest hexahexaflexagon, we were confounded to find that it was producing a strip of multicolored material. Further flexing of the hexahexaflexagon finally disgorged a gum-chewing stranger.

Unfortunately he was in a weak state and, owing to an apparent loss of memory, unable to give any account of how he came to be with us. His health has now been restored on our national diet of porridge, haggis and whisky, and he has become quite a pet around the department, answering to the name of Eccles.

Our problem is, should we now return him and, if so, by what method? Unfortunately Eccles now cringes at the very sight of a hexahexaflexagon and absolutely refuses to ‘flex.’

Saving Face

“The Joker,” a picture-preserving geomagic square by Lee Sallows. The 16 pieces can be assembled in varying groups of 4 to produce the same picture in 16 different ways, without rotation or reflection.

The outline need not be a joker — it can take almost any shape.


After seeing a manuscript of On the Origin of Species in April 1859, the Rev. Whitwell Elwin suggested that Darwin write about pigeons instead.

“This appears to me an admirable suggestion,” he wrote. “Everybody is interested in pigeons.”

Magic Variations

frolov pentagon

From Edward Falkener’s Games Ancient and Oriental and How to Play Them (1892), a magic pentagon. “It will be observed that the five sides of each pentagon are all equal, and that the five diameters, from one angle to the centre of the opposite side, are each 459, which is nine times the central number 51, which is also the mean number, the series being 1-101. And, further, that the inner pentagon is 510, or 10 times the mean number; the next pentagon 1,020, or 20 times the mean; the next 1,530, or 30 times the mean; and the outside pentagon 2,040, or 40 times the mean.” Evidently this was devised by Mikhail Frolov for Les carrés magiques (1886).

From the Gentleman’s Magazine, October 1768, a “magic circle of circles” by Benjamin Franklin:

The numbers run from 12 to 75. Each ring and each radius, added to the central number 12, gives 360, the number of degrees in a circle. The dashed lines define four additional sets of circles, with centers at A, B, C, and D, each with five rings; each ring, when added to 12, also gives the total 360. (The 12 is added arbitrarily to bring the total to 360; remove it and the whole arrangement remains magic.)

From the inimitable R.V. Heath:

heath cube

This 8×8 magic square can be cut into four smaller magic squares. When these four are stacked in the order indicated they produce a pandiagonal magic cube, each of whose rows, columns, and diagonals produces the total 130. Also: “The original eighth-order magic square has the additional property that if either set of alternate rows and either set of alternate columns be deleted — and this can be done in four ways — the remaining 16 numbers form a fourth-order magic square of magic constant 130.” (From Howard Whitley Eves, Mathematical Circles Squared, 1972.)

(Thanks, Ray.)

Still LifeÉcorché_cavalier_Fragonard_Alfort_1_edit1.jpg

French anatomist Honoré Fragonard (1732-1799) blurred the line between science and art by arranging human and animal bodies in fanciful poses. By replacing the eyeballs with glass replicas and injecting a distorting resin into the facial blood vessels, he achieved some remarkably expressive effects — his Fetus Dancing the Jig is best left to the imagination.

Florence’s Museum of Zoology and Natural History preserves a collection of wax models that were used in teaching medicine in the 18th century (below). Modelers might refer to 200 corpses in preparing a single wax figure. “If we succeeded in reproducing in wax all the marvels of our animal machine,” wrote director Felice Fontana, “we would no longer need to conduct dissections, and students, physicians, surgeons and artists would be able to find their desired models in a permanent, odor-free and incorruptible state.” Goethe praised artificial anatomy as “a worthy surrogate that, ideally, substitutes reality by giving it a hand.”

(From Roberta Panzanelli, ed., Ephemeral Bodies, 2008.)

Descartes’ Theorem

If three circles “kiss,” like the black circles above, then a fourth circle can be drawn that’s tangent to all three. In 1643 René Descartes showed that if the curvature or “bend” of a circle is defined as k = 1/r, then the radius of the fourth circle can be found by

descartes' theorem

The ± sign reflects the fact that two solutions are generally possible — the plus sign corresponds to the smaller red circle, the minus sign to the larger (circumscribing) one.

Frederick Soddy summed this up in a poem in Nature (June 20, 1936):

The Kiss Precise

For pairs of lips to kiss maybe
Involves no trigonometry.
‘Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

(Thanks, Sean.)

The Air Loom

In January 1797 London tea broker James Tilly Matthews was committed to the Bethlem psychiatric hospital after increasingly erratic outbursts in which he claimed he was being persecuted by political enemies. In 1809 Matthews’ friends petitioned for his release, arguing that he was no longer insane, and Bethlem apothecary John Haslam published a book-length study showing how bad his case had become.

Matthews believed that a gang of spies were occupying a Roman wall near the asylum and torturing him with a device called an air loom. The loom was operated by “the Middle Man,” while “Sir Archy” and the “Glove Woman” focused its rays on Matthews and “Jack the Schoolmaster” recorded their effects. Similar gangs, Matthews said, were operating looms all over London to influence the thinking of the nation’s leaders. The tortures included “fluid locking,” “cutting soul from sense,” “stone making,” “thigh talking,” and “lobster-cracking,” which Matthews also described as “sudden death-squeezing”:

In short, I do not know any better way for a person to comprehend the general nature of such lobster-cracking operation, than by supposing himself in a sufficiently large pair of nut-crackers or lobster-crackers, with teeth which should pierce as well as press him through every particle within and without; he experiencing the whole stress, torture, driving, oppressing, and crush all together.

Matthews remained an asylum inpatient until his death in 1815. His is now considered to be the first documented case of paranoid schizophrenia.


The programming language Chef, devised by David Morgan-Mar, is designed to make programs look like cooking recipes. Variables are represented by “ingredients,” input comes from the “refrigerator,” output is sent to “baking dishes,” and so on. The language’s design principles state that “program recipes should not only generate valid output, but be easy to prepare and delicious,” but many of them fall short of that goal — one program for soufflé correctly prints the words “Hello world!”, but the recipe requires 32 zucchinis, 101 eggs, and 111 cups of oil to be combined in a bowl and served to a single person. Mike Worth set out to write a working program that could also be read as an actual recipe. Here’s what he came up with:

Hello World Cake with Chocolate sauce.

This prints hello world, while being tastier than Hello World Souffle. The main
chef makes a " world!" cake, which he puts in the baking dish. When he gets the
sous chef to make the "Hello" chocolate sauce, it gets put into the baking dish
and then the whole thing is printed when he refrigerates the sauce. When
actually cooking, I'm interpreting the chocolate sauce baking dish to be
separate from the cake one and Liquify to mean either melt or blend depending on

33 g chocolate chips
100 g butter
54 ml double cream
2 pinches baking powder
114 g sugar
111 ml beaten eggs
119 g flour
32 g cocoa powder
0 g cake mixture

Cooking time: 25 minutes.

Pre-heat oven to 180 degrees Celsius.

Put chocolate chips into the mixing bowl.
Put butter into the mixing bowl.
Put sugar into the mixing bowl.
Put beaten eggs into the mixing bowl.
Put flour into the mixing bowl.
Put baking powder into the mixing bowl.
Put cocoa  powder into the mixing bowl.
Stir the mixing bowl for 1 minute.
Combine double cream into the mixing bowl.
Stir the mixing bowl for 4 minutes.
Liquify the contents of the mixing bowl.
Pour contents of the mixing bowl into the baking dish.
bake the cake mixture.
Wait until baked.
Serve with chocolate sauce.

chocolate sauce.

111 g sugar
108 ml hot water
108 ml heated double cream
101 g dark chocolate
72 g milk chocolate

Clean the mixing bowl.
Put sugar into the mixing bowl.
Put hot water into the mixing bowl.
Put heated double cream into the mixing bowl.
dissolve the sugar.
agitate the sugar until dissolved.
Liquify the dark chocolate.
Put dark chocolate into the mixing bowl.
Liquify the milk chocolate.
Put milk chocolate into the mixing bowl.
Liquify contents of the mixing bowl.
Pour contents of the mixing bowl into the baking dish.
Refrigerate for 1 hour.

Worth confirmed that this correctly prints the words “Hello world!”, and then he used the same instructions to bake a real cake. “It was surprisingly well received,” he writes. “The cake was slightly dry (although nowhere near as dry as cheap supermarket cakes), but this was complimented well by the sauce. My brother even asked me for the recipe!”

While we’re at it: Fibonacci Numbers With Caramel Sauce.

Sweet Story

In 1987, paleontologist Tom Rich was leading a dig at Dinosaur Cove southwest of Melbourne when student Helen Wilson asked him what reward she’d get if she found a dinosaur jaw. He said he’d give her a kilo (2.2 pounds) of chocolate. She did, and he did.

Encouraged, the students asked Rich what they’d get if they found a mammal bone. These are fairly rare among dinosaur fossils in Australia, so Rich rashly promised a cubic meter of chocolate — 35 cubic feet, or about a ton.

The cove was “dug out” by 1994, and paleontologists shut down the dig. Rich sent a curious unclassified bone, perhaps a turtle humerus, to two colleagues, who recognized it as belonging to an early echidna, or spiny anteater — a mammal.

Rich now owed the students $10,000 worth of chocolate. “It turns out that it is technically impossible to make a cubic meter of chocolate, because the center would never solidify,” he told National Geographic in 2005. So he arranged for a local Cadbury factory to make a cubic meter of cocoa butter, and then turned the students loose in a room full of chocolate bars.

“It was a bit like Willy Wonka,” Wilson said. “There were chocolate bars on the counters, the tables. We carried out boxes and boxes of chocolate.”

Fittingly, the new echidna was named Kryoryctes cadburyi.

Wheels of Justice

In 1962 a Swedish motorist was fined for leaving his car too long in a space with a posted time limit. The motorist objected, saying that he had removed the car in time and then happened to return to the same spot later, resetting the time limit. The policeman defended his charge, saying that he had noted the positions of the valves on two of the tires — the front-wheel valve was in the 1 o’clock position, the rear-wheel valve at 8 o’clock. If the car had been moved, he argued, the valves were unlikely to take the same positions.

The court accepted the motorist’s claim, calculating that the chance that the valves would return to the same positions by chance was 1/12 × 1/12 = 1/144, great enough to establish reasonable doubt. The court added that if all four valves had been found to be in the same position, the lower likelihood (1/12 × 1/12 × 1/12 × 1/12 = 1/20,736, it figured) would have been enough to uphold the fine.

Is this right? In evaluating this reasoning, University of Chicago law professor Hans Zeisel notes that this method is biased in favor of the defendant, since the positions of the valves are not perfectly independent. He later added, “The use of the 1/144 figure for the probability of the constable’s observations on the assumption that the defendant had driven away also can be questioned. Not only may the rotations of the tires on different axles be correlated, but the figure overlooks the observation that the car was in the same parking spot. When a person leaves a parking place, it is far from certain that the spot will be available later and that the person will choose it again. For this reason, it has been said that the probability of a coincidence is even smaller than a probability involving only the valves.”

(Hans Zeisel, “Dr. Spock and the Case of the Vanishing Women Jurors,” University of Chicago Law Review, 37:1 [Autumn 1969], 1-18)

Shadow Play

I am watching a double solar eclipse. The heavenly body Far, traveling east, passes before the sun. Beneath it passes the smaller body Near, traveling west. Far and Near appear to be the same size from my vantage point. Which do I see?

Common sense says that I see Near, since it’s closer. But Washington University philosopher Roy Sorensen argues that in fact I see Far. Near’s existence has no effect on the pattern of light that reaches my eyes. It’s not a cause of what I’m seeing; the view would be the same without it. (Imagine, for example, that Far were much larger and Near was lost in its shadow.)

“When objects are back-lit and are seen by virtue of their silhouettes, the principles of occlusion are reversed,” Sorensen concludes. “In back-lit conditions, I can hide a small suitcase by placing a large suitcase behind it.”

See In the Dark.

(Roy Sorensen, “Seeing Intersecting Eclipses,” Journal of Philosophy XCVI, 1 (1999): 25-49.)

World View

In Other Inquisitions, Borges writes of a strange taxonomy in an ancient Chinese encyclopedia:

On those remote pages it is written that animals are divided into (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained, (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g), stray dogs, (h) those that are included in this classification, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very fine camel’s hair brush, (l) others, (m) those that have just broken a flower vase, (n) those that resemble flies from a distance.

This is fanciful, but it has the ring of truth — different cultures can classify the world in surprisingly different ways. In traditional Dyirbal, an aboriginal language of Australia, each noun must be preceded by a variant of one of four words that classify all objects in the universe:

  • bayi: men, kangaroos, possums, bats, most snakes, most fishes, some birds, most insects, the moon, storms, rainbows, boomerangs, some spears, etc.
  • balan: women, bandicoots, dogs, platypus, echidna, some snakes, some fishes, most birds, fireflies, scorpions, crickets, the hairy mary grub, anything connected with water or fire, sun and stars, shields, some spears, some trees, etc.
  • balam: all edible fruit and the plants that bear them, tubers, ferns, honey, cigarettes, wine, cake
  • bala: parts of the body, meat, bees, wind, yamsticks, some spears, most trees, grass, mud, stones, noises and language, etc.

“The fact is that people around the world categorize things in ways that both boggle the Western mind and stump Western linguists and anthropologists,” writes UC-Berkeley linguist George Lakoff in Women, Fire, and Dangerous Things (1987). “More often than not, the linguist or anthropologist just throws up his hands and resorts to giving a list — a list that one would not be surprised to find in the writings of Borges.”

In a Word

adj. producing fever

The 1895 meeting of the Association of American Physicians saw a sobering report: Abraham Jacobi presented the case of a young man whose temperature had reached 149 degrees.

Nonsense, objected William Henry Welch. Such an observation was impossible. He recalled a similar report in the Journal of the American Medical Association (March 31, 1891) in which a Dr. Galbraith of Omaha had found a temperature of 171 degrees in a young woman.

“I do not undertake to explain in what way deception was practised, but there is no doubt in my mind that there was deception,” he said. “Such temperatures as those recorded in Dr. Galbraith’s and Dr. Jacobi’s cases are far above the temperature of heat rigor of mammalian muscle, and are destructive of the life of animal cells.”

Jacobi defended himself: Perhaps medicine simply hadn’t developed a theory to account for such things. But another physician told Welch that Galbraith’s case at least had a perfectly satisfactory explanation — another doctor had caught her in “the old-fashioned trick of heating the thermometer by a hot bottle in the bed.”

Self-Tiling Tile Sets

These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):

sallows rep-tiles 1

The larger versions could then perform the same trick, and so on. Here’s another set:

sallows rep-tiles 2

In this set, each of the six pieces is paved by some four of them:

sallows rep-tiles 3

By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.


Here’s a way to visualize multiplication that reduces it to simple counting:

multiplication lattice

Express the digits in each factor with rows of parallel lines, as shown, and then count the intersections to derive the product. This is more cumbersome than the traditional method, but its visual nature is appealing, and it permits anyone who can count to reach the right answer even if he doesn’t know the multiplication table.

The example above uses small digits, so no “carrying” is required, but the method does accommodate more complex sums — it’s explained well in this video:

See Two by Two.

(Thanks, Dieter.)


A card trick by Mark Wilson:

wilson card trick

Put your finger on any red card. Move it left or right to the nearest black card. Move it up or down to the nearest red card. Move it diagonally to the nearest black card. Now move it down or to the right to the nearest red card.

You’ll always land on the ace of diamonds.

(From Harold R. Jacobs, Mathematics: A Human Endeavor, 1970.)

A Geometric Paradox

a geometric paradox

“A Positive Reminder”

A carpenter named Charlie Bratticks,
Who had a taste for mathematics,
One summer Tuesday, just for fun,
Made a wooden cube side minus one.

Though this to you may well seem wrong,
He made it minus one foot long,
Which meant (I hope your brains aren’t frothing)
Its length was one foot less than nothing,

Its width the same (you’re not asleep?)
And likewise minus one foot deep;
Giving, when multiplied (be solemn!),
Minus one cubic foot of volume.

With sweating brow this cube he sawed
Through areas of solid board;
For though each cut had minus length,
Minus times minus sapped his strength.

A second cube he made, but thus:
This time each one-foot length was plus:
Meaning of course that here one put
For volume, plus one cubic foot.

So now he had, just for his sins,
Two cubes as like as deviant twins:
And feeling one should know the worst,
He placed the second in the first.

One plus, one minus — there’s no doubt
The edges simply canceled out;
So did the volume, nothing gained;
Only the surfaces remained.

Well may you open wide your eyes,
For those were now of double size,
On something which, thanks to his skill,
Took up no room and measured nil.

From solid ebony he’d cut
These bulky cubic objects, but
All that remained was now a thin
Black sharply-angled sort of skin

Of twelve square feet — which though not small,
Weighed nothing, filled no space at all.
It stands there yet on Charlie’s floor;
He can’t think what to use it for!

— J.A. Lindon

All the News …

How’s that for a headline? It ran in the New York Times Sunday magazine on Aug. 27, 1911:

Canals a thousand miles long and twenty miles wide are simply beyond our comprehension. Even though we are aware of the fact that … a rock which here weighs one hundred pounds would there only weigh thirty-eight pounds, engineering operations being in consequence less arduous than here, yet we can scarcely imagine the inhabitants of Mars capable of accomplishing this Herculean task within the short interval of two years.

The Times was relying on Percival Lowell, who was convinced that a dying Martian civilization was struggling to reach the planet’s ice caps. “The whole thing is wonderfully clear-cut,” he’d told the newspaper — but he was already largely ostracized by skeptical colleagues who couldn’t duplicate his findings. The “spokes” he later saw on Venus may have been blood vessels in his own eye.

Whatever his shortcomings, Lowell’s passions led to some significant accomplishments, including Lowell Observatory and the discovery of Pluto 14 years after his death. “Science,” wrote Emerson, “does not know its debt to imagination.”


A self-reproducing sentence by Lee Sallows — “Doing what it tells you to do yields a replica of itself”:

sallows palingram

This reminds me of a short short story by Fredric Brown:


Professor Jones had been working on time theory for many years.

“And I have found the key equation,” he told his daughter one day. “Time is a field. This machine I have made can manipulate, even reverse, that field.”

Pushing a button as he spoke, he said, “This should make time run backward run time make should this,” said he, spoke he as button a pushing.

“Field that, reverse even, manipulate can made have I machine this. Field a is time.” Day one daughter his told he, “Equation key the found have I and.”

Years many for theory time on working been had Jones Professor.


Second Chances

In science it often happens that scientists say, ‘You know, that’s a really good argument; my position is mistaken,’ and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn’t happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.

— Carl Sagan, in a 1987 address, quoted in Jon Fripp et al., Speaking of Science, 2000

To Whom It May Concern

On Jan. 9, 1793, two astonished farmers in Woodbury, N.J., watched a strange craft descend from the sky into their field. An excited Frenchman greeted them in broken English and gave them swigs of wine from a bottle. Unable to make himself understood, he finally presented a document:

The farmers helped the man fold his craft and load it onto a wagon for the trip back to Philadelphia. Before leaving, the Frenchman asked them to certify the time and place of his arrival. These details were important — he was Jean-Pierre Blanchard, and he had just completed the first balloon flight in North America.

One hundred sixty-nine years later, when John Glenn went into orbit aboard Friendship 7 in 1962, mission planners weren’t certain where he’d come down. The most likely sites were Australia, the Atlantic Ocean, and New Guinea, but it might be 72 hours before he could be picked up.

Glenn worried about spending three days among aborigines who had seen a silver man emerge from “a big parachute with a little capsule on the end,” so he took with him a short speech rendered phonetically in several languages. It read:

“I am a stranger. I come in peace. Take me to your leader, and there will be a massive reward for you in eternity.”

Three Theorems

Theorem 1. A crocodile is longer than it is wide.

Proof. A crocodile is long on the top and bottom, but it is green only on the top; therefore a crocodile is longer than it is green. A crocodile is green along both its length and width, but it is wide only along its width; hence a crocodile is greener than it is wide. Therefore a crocodile is longer than it is wide.

Theorem 2. Napoleon was a poor general.

Proof. Most men have an even number of arms. Napoleon was warned that Wellington would meet him at Waterloo. To be forewarned is to be forearmed. But four arms is a very odd number of arms for a man. The only number that is both even and odd is infinity. Therefore, Napoleon had an infinite number of arms in his battle against Wellington. A general who loses a battle despite having an infinite number of arms is very poor general.

Theorem 3. If 1/0 = ∞, then 1/∞ = 0.

Proof. Given:

three theorems 2

Rotate both sides 90° counterclockwise:

three theorems 3

Subtract 8 from both sides:

three theorems 4

Now reverse the rotation:

three theorems 5

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.)