Science & Math

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.)

Working Afield

sigurya barbulata

Dutch author Leo Lionni devoted most of his career to children’s books, but in 1977 he undertook a weird experiment. Parallel Botany is a catalog of made-up plants, whose made-up features are described by made-up botanists and illustrated by Lionni’s pencil drawings. Sigurya barbulata, at left, is distinguished by its crowning “cephalocarpus”; a specimen discovered in a Mexican pyramid was found to have been metallized into an organic mace, but how this had come about is the subject of “furious debates.”

“The difficulties of applying traditional methods of research to the study of parallel botany stem chiefly from the matterlessness of the plants,” Lionni wrote. “Deprived as they are of any real organs or tissues, their character would be completely indefinable if it were not for the fact that parallel botany is nonetheless botany, and as such it reflects, even if somewhat distantly, many of the most evident features of normal plants.”

Why do all this? Lionni closes with a quote by the made-up Swedish philosopher Erud Kronengaard: “There are two kinds of men, those who are capable of wonder and those who are not. I hope to God that it is the first who will forge our destiny.”

Alison’s Triangle

alison's triangle

I’m not sure who came up with this — this simple diagram reflects all possible true trigonometric identities of the form x ÷ y = z or x × y = z, where x, y, and z are the basic trigonometric functions of the same angle t.

For any three neighboring functions on the perimeter of the star, the product of the ends always equals the middle (e.g., tan t × cos t = sin t) and the middle function divided by one of the end functions is equal to the other end function (e.g., sin t ÷ tan t = cos t and sin t ÷ cos t = tan t). If you memorize the diagram you can reel off a list of 18 simple relations.

I found it in Michael Stueben’s Twenty Years Before the Blackboard, 1998.

Shape Reference

thomas whales

The index to the fourth edition of George Thomas’ Calculus and Analytic Geometry contains an entry for “Whales” on page 188. That page contains no reference to whales, but it does include the figure above.,Erich_1920_G%C3%B6ttingen.jpg?uselang=de

German mathematician Erich Bessel-Hagen was often teased for his protruding ears.

In 1923 his colleague Béla Kerékjártó published a book, Vorlesungen Über Topologie, whose index lists a reference to Bessel-Hagen on page 151.

That page makes no mention of Bessel-Hagen, but it does contain this figure:


Is that libel?

Corner Market

corner market diagram

From Martin Gardner, via Michael Stueben: Obtain a slab of gold measuring 10″ x 11″ x 1″. Divide it diagonally and then cut a triangular notch in two corners as shown. Remove these notches as profit, and slide the remaining halves together to produce a new 10″ x 11″ x 1″ slab. The process can be repeated to yield any amount of money you like!

No Attraction

Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.

If gravity were turned off, would this still be true?

Click for Answer


Pretend that you’ve never seen this before and that it’s an actual living person whose personality you’re trying to read. If you look directly at her face, she seems to hesitate, but if you look near it, say beyond her at the landscape, and try to sense her mood, she smiles at you.

In studying this systematically, Harvard neurobiologist Margaret Livingstone found that “if you look at this painting so that your center of gaze falls on the background or her hands, Mona Lisa’s mouth — which is then seen by your peripheral, low-resolution, vision — appears much more cheerful than when you look directly at it, when it is seen by your fine-detail fovea.

“This explains its elusive quality — you literally can’t catch her smile by looking at it. Every time you look directly at her mouth, her smile disappears because your central vision does not perceive coarse image components very well. People don’t realize this because most of us are not aware of how we move our eyes around or that our peripheral vision is able to see some things better than our central vision. Mona Lisa smiles until you look at her mouth, and then her smile fades, like a dim star that disappears when you look directly at it.”

(From her book Vision and Art: The Biology of Seeing, 2002.)

Worth Remembering

William D. Harvey offered this in Omni in 1980 — a mnemonic for spelling mnemonics:

Mnemonics neatly eliminate man’s only nemesis: insufficient cerebral storage.

First Things First

We [Einstein and Ernst Straus] had finished the preparation of a paper and were looking for a paper clip. After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more drawers we came upon a whole box of unused paper clips. Einstein immediately started to shape one of them into a tool to straighten the bent one. When asked what he was doing, he said, ‘Once I am set on a goal, it becomes difficult to deflect me.’

– Ernst Straus, “Memoir,” in A.P. French, ed., Einstein: A Centenary Volume, 1979

(Einstein said to an assistant at Princeton that this was the most characteristic anecdote that could be told of him.)


“[John] von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of von Neumann’s advice. It’s made me a very happy man ever since. But it was von Neumann who put the seed in that grew into my active irresponsibility!” — Richard Feynman

He expands on this in Christopher Sykes’ No Ordinary Genius (1994):

“I got the idea of ‘active irresponsibility’ in Los Alamos. We often went on walks, and one day I was with the great mathematician von Neumann and a few other people. I think Bethe and von Neumann were discussing some social problem that Bethe was very worried about. Von Neumann said, ‘I don’t feel any responsibility for all these social problems. Why should I? I’m born into the world, I didn’t make it.’ Something like that. Well, I’ve read von Neumann’s autobiography and it seems to me that he felt perpetually responsible, but at that moment this was a new idea to me, and I caught onto it. Around you all the time there are people telling you what your responsibilities are, and I thought it was kind of brave to be actively irresponsible. ‘Active’ because, like democracy, it takes eternal vigilance to maintain it — in a university you have to perpetually watch out, and be careful that you don’t do anything to help anybody!”

Hans Bethe:

“Feynman somehow was proud of being irresponsible. He concentrated on his science, and on enjoying life. There are some of us — including myself — who felt after the end of the Second World War that we had a great responsibility to explain atomic weapons, and to try and make the government do sensible things about atomic weapons. … Feynman didn’t want to have anything to do with it, and I think quite rightly. I think it would be quite wrong if all scientists worked on discharging their responsibility. You need some number of them, but it should only be a small fraction of the total number of scientists. Among the leading scientists, there should be some who do not feel responsible, and who only do what science is supposed to accomplish.”

Marvin Minsky:

“I must say I have a little of this sense of social irresponsibility, and Feynman was a great inspiration to me — I have done a good deal of it since. There are several reasons for a scientist to be irresponsible, and one of them I take very seriously: people say, ‘Are you sure you should be working on this? Can’t it be used for bad?’ Well, I have a strong feeling that good and bad are things to be thought about by people who understand better than I do the interactions among people, and the causes of suffering. The worst thing I can imagine is for somebody to ask me to decide whether a certain innovation is good or bad.”

Langton’s Ant

Set an ant down on a grid of squares and ask it to follow two rules:

  1. If you find yourself on a white square, turn 90° right, change the color of the square to black, and move forward one unit.
  2. If you find yourself on a black square, turn 90° left, change the color of the square to white, and move forward one unit.

That’s it. At first the ant will seem to mill around uncertainly, as above, producing an irregular jumble of black and white squares. But after about 10,000 steps it will start to build a “highway,” following a repeating loop of 104 steps that unfolds forever (below). Computer scientist Chris Langton discovered the phenomenon in 1986.

Will this happen even if some of the starting squares are black? So far the answer appears to be yes — in every initial configuration that’s been tested, the ant eventually produces a highway. If there’s an exception, no one has found it yet.