Shadow Play

http://commons.wikimedia.org/wiki/File:Ombra_astrofilo.png
Image: Wikimedia Commons

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.)

| Science & Math

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.”

| Language · Literature · Science & Math

In a Word

febrifacient
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.”

| Language · Science & Math

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.

| Science & Math

Intersections

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.)

| Science & Math

Presto

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.)

| Science & Math

“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

| Poems · Science & Math

All the News …

http://query.nytimes.com/gst/abstract.html?res=9D04E2DE163CE633A25754C2A96E9C946096D6CF

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.”

| Oddities · Science & Math

Palingram

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:

THE END

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.

END THE

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