Science & Math

One More Try

http://books.google.com/books?id=FjYDAAAAQAAJ&printsec=frontcover&rview=1#v=onepage&q&f=false

A perpetual-motion scheme by William Congreve. A, B, and C are three horizontal rollers fixed in a frame. They’re surrounded by a continuous band of sponge, a, and that’s surrounded by a chain of weights, b. Immerse the whole thing partially in a cistern. The sponge on the left will absorb water by capillary action, say from x to y; the sponge on the right will not (because the weights squeeze it out!). “The band will begin to move in the direction A B; and as it moves downwards, the accumulation of water will continue to rise, and thereby carry on a constant motion.”

From Henry Dircks, Perpetuum Mobile; or, Search for Self-Motive Power, 1861.

Can Do

http://books.google.com/books?id=dOtLAAAAIAAJ&source=gbs_navlinks_s

In 1921, chemists at Arthur D. Little Inc. reduced 100 pounds of sows’ ears to glue, converted it to gelatin, forced it into fine strands, and wove these into a purse “of the sort which ladies of great estate carried in medieval days — their gold coin in one end and their silver coin in the other.”

“We made this silk purse from a sow’s ear because we wanted to, because it might serve as an example to clients who come to us with their ambitions or their troubles, and also as a contribution to philosophy,” they reported. “Things that everybody thinks he knows only because he has learned the words that say it, are poisons to progress.”

Canceling Zeros

canceling zeros

Misc

  • Christopher Lee is Ian Fleming’s cousin.
  • £12.12s.8d = 12128 farthings
  • ii is real.
  • Shouldn’t Juliet have asked, “Wherefore art thou Montague?”
  • “Of soup and love, the first is the best.” — Thomas Fuller

Speed Limits

http://commons.wikimedia.org/wiki/File:Brehms_Het_Leven_der_Dieren_Zoogdieren_Orde_4_Afrikaanse_Gepard_(Cynailurus_guttatus).jpg

The cheetah can reach speeds over 70 mph. In a dive, the peregrine falcon can reach 200 mph. But in 1927, entomologist Charles Townsend estimated that the deer botflies he’d observed in New Mexico surpassed both of these, reaching 400 yards per second. That’s 818 mph.

This claim stood for 11 years, until in 1938 chemist Irving Langmuir debunked it in Science:

  • The power needed to maintain this speed amounts to 370 watts, or about half a horsepower. To deliver it, the fly would have to consume 1.5 times its own weight in food every second.
  • Ballistics formulas show that the wind pressure on the fly’s head would amount to 8 pounds per square inch, probably enough to crush the fly.
  • An 800 mph fly would strike the skin with a force of 310 pounds. “It is obvious that such a projectile would penetrate deeply into human flesh.”
  • A supersonic fly would be invisible to the eye, not the “brownish blur” that Townsend had described.

Not to mention that an 800 mph fly would create its own sonic boom. After weighing the facts, Langmuir concluded, “The description given by Dr. Townsend of the appearance of the flies seems to correspond best with a speed in the neighborhood of 25 m/hr.”

Space Bills

When an explosion crippled Apollo 13’s command module, the crew used the lunar module as a “lifeboat.” The two modules had been built by different contractors, so when the mission was over Grumman sent a tongue-in-cheek bill to Rockwell for “towing” the ship to the moon and back:

http://www.futilitycloset.com/2011/05/02/business-trip/

The Associated Press reported that “North American Rockwell replied that the invoice had been examined by the company’s auditor, who pointed out that North American Rockwell had not yet received payment for ferrying LMs to the moon on previous missions.”

(Thanks, Perry.)

The Monster Study

In 1939, University of Iowa graduate student Mary Tudor began an experiment with local orphans, warning them that they were showing signs of stuttering and lecturing them whenever they repeated a word. The children became acutely self-conscious, and many began to stutter, fulfilling the theory that “the affliction is caused by the diagnosis.”

Sixty years later, when Tudor was 84, she received a letter from one of the orphans. It was addressed to “Mary Tudor Jacobs The Monster.”

“You destroyed my life,” it ran. “I could have been a scientist, archaeologist or even president. Instead I became a pitiful stutterer. The kids made fun of me, my grades fell off, I felt stupid. Clear into my adulthood, I still want to avoid people to this day.”

“I didn’t like what I was doing to those children,” Tudor told the San Jose Mercury News in 2001. “It was a hard, terrible thing. Today, I probably would have challenged it. Back then you did what you were told. It was an assignment. And I did it.”

Rigged Latin

When entomologist Paul Marsh was given the chance to name two wasp species in the genus Heerz, he called them Heerz tooya and Heerz lukenatcha.

The International Commission on Zoological Nomenclature insists that “A zoologist should not propose a name that, when spoken, suggests a bizarre, comical, or otherwise objectionable meaning.” But a few get through. Examples:

  • Vini vidivici (parrot)
  • Apopyllus now (spider)
  • Lalapa lusa (wasp)
  • Agra vation (beetle)
  • Phthiria relativitae (bee fly)
  • Pison eyvae (wasp)
  • Ba humbugi (snail)
  • Eubetia bigaulae (“you betcha by golly”) (moth)

Three mythicomyiid flies are named Pieza kake, Pieza pi, and Pieza deresistans.

In 1912 the Zoological Society of London criticized entomologist George Kirkaldy for giving a series of hemipterans the generic names Polychisme, Elachisme, Marichisme, Dolichisme, Florichisme, and Ochisme (“Polly kiss me,” “Ella kiss me,” “Mary kiss me,” “Dolly kiss me,” “Flora kiss me,” “Oh, kiss me!”). In the same spirit, in 2002 a hopeful Neal Evenhuis named a fossil mythicomyiid Carmenelectra shechisme. “The offer’s still good,” he told the Chicago Tribune in 2008. “I’ll be willing to meet her.”

Head Games

The present King of France is bald.

Is this statement true or false? Well, it’s not true — France has no king presently. But if it’s false then its negation ought to seem true: The present King of France is not bald. That’s no better. Yet it’s not gibberish — the sentence seems to have a clear meaning that we can understand.

Bertrand Russell and Alfred North Whitehead spent much of their time at Cambridge debating this point. “It is astonishing what intricate and remote considerations can be brought to bear on this interesting question,” Russell wrote to his wife. “We finally decided that he isn’t, altho’ he has no hair of his own. Experienced people will infer that he wears a wig, but this would be a mistake.”

The Bouba/Kiki Effect

In 2001, USCD psychologist Vilayanur Ramachandran presented these shapes to American undergraduates and to Tamil speakers in India. He asked, “Which of these shapes is bouba and which is kiki?” Around 98% of the respondents assigned the name kiki to the spiky shape and bouba to the curvy one.

Psychologist Wolfgang Köhler had found a similar effect in 1929 using the words baluba and takete. “This result suggests that the human brain is somehow able to to extract abstract properties from the shapes and sounds,” Ramachandran wrote, “for example, the property of jaggedness embodied in both the pointy drawing and the harsh sound of kiki.”

The Paradox of Choices

http://commons.wikimedia.org/wiki/File:Illustration_at_page_288_in_Grimm%27s_Household_Tales_(Edwardes,_Bell).png

When you’re a princess you have to kiss a lot of frogs. But you can see only one frog at a time, and once you reject a frog you can’t return to it. How can you know when to stop hoping for a prince and settle down with the frog you have?

“Surprisingly, … there is a method which enables us to select the best candidate with a probability of nearly 30% even if n is a large number,” writes Gabor Szekely in Paradoxes in Probability Theory and Mathematical Statistics (2001). “Let the first 37% (more precisely, 100/e%) of the candidates go and then select the first one better than any previous candidate (if none are better, select the last). In this case the chance of selecting the best is approximately 1/e, i.e. ≈37% however great n is.”

So if there are 100 frogs in the forest, reject the first 37 and then choose the first one that seems to beat all the others. There’s about a 37 percent chance that it’s the best one.

Antipodes

What is directly on the opposite side of the world from you? This map answers that question by superimposing each point on earth with its opposite. Some notable sisters:

  • Beijing, China, is nearly opposite Buenos Aires, Argentina
  • Madrid, Spain, is nearly opposite Wellington, New Zealand
  • Bogotá, Colombia, is nearly opposite Jakarta, Indonesia
  • Bangkok, Thailand, is nearly opposite Lima, Peru
  • Quito, Ecuador, is nearly opposite Singapore
  • Seoul, South Korea, is nearly opposite Montevideo, Uruguay
  • Perth, Australia, is nearly opposite Bermuda
  • Charmingly, Cherbourg, France, is opposite the Antipodes Islands south of New Zealand

W.V.O. Quine explains how an enterprising traveler can arrange to visit two precise antipodes: “Note to begin with that any route from New York to Los Angeles, if not excessively devious, is bound to intersect any route from Winnipeg to New Orleans. Now let someone travel from New York to Los Angeles, and also travel from roughly the antipodes of Winnipeg to roughly the antipodes of New Orleans. These two routes do not intersect — far from it; but one of them intersects a route that is antipodal to the other. So our traveler is assured of having touched a pair of mutually antipodal points precisely, though he will know only approximately where.”

Dueling Expectations

In 2007, New Scientist announced that the best strategy in a game of rock paper scissors is to choose scissors.

Research has shown that rock is the most popular of the three moves. If your opponent expects you to choose it, he’ll choose paper in order to beat it — in which case scissors will win.

In 2005 a Japanese art collector asked Christie’s and Sotheby’s to play a match, saying the winner could sell his impressionist paintings. The 11-year-old daughter of a Christie’s director recommended scissors, saying, “Everybody expects you to choose rock.”

Sure enough, Sotheby’s chose paper, and Christie’s won the £10 million deal.

Riddle

If there was a time when nothing existed, then there must have been a time before that — when even nothing did not exist. Suddenly, when nothing came into existence, could one really say whether it belonged to the category of existence or of non-existence?

— Chuang-Tzu

The Hairy Ball Theorem

http://commons.wikimedia.org/wiki/File:Hairy_ball.png

A hairy ball can’t be combed flat — it must always have a cowlick.

This result arose originally in algebraic topology, but it has intriguing applications elsewhere. For example, it can’t be windy everywhere at once on Earth’s surface — at any given moment, the horizontal wind speed somewhere must be zero.

Perpetual Locomotion

http://books.google.com/books?id=W5A5AAAAcAAJ&printsec=frontcover#v=onepage&q&f=false

In 1829 a correspondent to the Mechanic’s Magazine proposed this design for a “self-moving railway carriage.” Fill the car with passengers and cargo as shown and set it on two rails that undulate across the landscape:

http://books.google.com/books?id=19VQAAAAYAAJ&printsec=frontcover#v=onepage&q&f=false

In the descending sections (a, c, e) the two rails are parallel. In the ascending ones (b, d) they diverge so that the car, mounted on cones, will roll forward to settle more deeply between them, paradoxically “ascending” the slope. If the track circles the world the car will “assuredly continue to roll along in one undeviating course until time shall be no more.”

“How any one could ever imagine that such a contrivance would ever continue in motion for even a short time … must be a puzzle to every sane mechanic,” wrote John Phin in The Seven Follies of Science in 1911. But what does he know?

Business Trip

http://commons.wikimedia.org/wiki/File:Michael_collins_training_apollo_11.jpg

As a joke, Michael Collins submitted a travel voucher for his trip aboard Gemini 10. NASA reimbursed him $8 per day, a total of $24.

In his autobiography, Collins notes that he could instead have claimed 7 cents a mile, which would have yielded $80,000.

But one of the original Mercury astronauts had already tried this — and had received a bill for “a couple of million dollars” for the rocket he’d used.

Eye to Eye

equal chord theorem

Draw two circles of any size and bracket them with tangents, as shown.

The chords in green will always be equal.

The Elevator Problem

Any group of six people must contain at least three mutual friends or three mutual strangers.

Represent the people with dots, and connect friends with blue lines and strangers with red. Will the completed diagram always contain a red or a blue triangle?

Because A has five relationships and we’re using two colors, at least three of A’s connections must be of the same color. Say they’re friends:

elevator problem 1

Already we’re perilously close to completing a triangle. We can avoid doing so only if B, C, and D are mutual strangers — in which case they themselves complete a triangle:

elevator problem 2

We can reverse the colors if B, C, and D are strangers to A, but then we’ll get the complementary result. The completed diagram must always contain at least one red or blue triangle.

I think this problem appeared originally in the William Lowell Putnam mathematics competition of 1953. Six is the smallest number that requires this result — a group of five people would form a pentagon in which the perimeter might be of one color and the internal connections of another.

(Update: In fact the more general version of this idea was adduced in 1930 by Cambridge mathematician F.P. Ramsey. It is very interesting.) (Thanks, Alex.)

Holditch’s Theorem

http://books.google.com/books?id=978KAAAAYAAJ&source=gbs_navlinks_s

Cut a notch in a stick and label the two parts p and q. Then draw the stick around the shore of a pond. The notch will describe a curve, and, remarkably, the area between the shore and this curve will be given by πpq.

“Two things immediately struck me as astonishing,” wrote British mathematician Mark Cooker in 1988. “First, the formula for the area is independent of the size of the given curve. Second, [the equation for the area] is the area of an ellipse of semi-axes p and q, but there are no ellipses in the theorem!”

Knot Mirage

For years Raymond Smullyan sought a “metaparadox,” a statement that is paradoxical if and only if it isn’t. He arrived at this:

Either this sentence is false, or (this sentence is paradoxical if and only if it isn’t).

He wrote, “I leave the proof to the reader.”

Top Secret

In 1976, CUNY mathematician Robert Feinerman showed that the game of dreidel is fundamentally unfair.

Each player contributes one unit to the pot, and then all take turns spinning the top. Each spin produces one of four outcomes: the player does nothing, collects the entire pot, collects half the pot, or puts one unit into the pot. When a player collects the entire pot, then each player contributes one unit to form a new pot and play continues.

Feinerman found that, if Xn is the payoff on the nth spin and p is the number of players, the expected value of Xn is

dreidel expected value

Thus if there are more than two players, “the first player has an unfair advantage over the second player, who in turn has an unfair advantage over the third player, etc.”

Dreidel is booming nonetheless. A Major League Dreidel tournament has been held in New York City every Hanukkah since 2007. The official playing surface is called the Spinagogue, and the tournament slogan is “no gelt, no glory.”

(Robert Feinerman, “An Ancient Unfair Game,” American Mathematical Monthly 83:623-625.)

Cancel That

freak cancellations

A Feathered Maître d’

The greater honeyguide of Africa eats beeswax but isn’t always able to invade a hive on its own. So it has forged a unique partnership with human beings: The bird attracts the attention of local honey hunters with a chattering call, flies toward a hive, then stops and calls again. When they arrive at the hive, the humans open it, subdue the bees with smoke, take the honey, and leave the wax for the bird.

This arrangement saves the humans an average of 5.7 hours in searching for hives, but it’s not foolproof. “We have been ‘guided’ to an abrupt precipice and to a bull elephant by greater honeyguides,” report biologists Lester Short and Jennifer Horne. “In these cases there were bee-hives below the cliff (in a valley) and beyond the elephant. Concern for the welfare of the guided person is beyond any reasonable expectation of a honeyguide.”

(Thanks, Tom.)