The Pitot Theorem

http://commons.wikimedia.org/wiki/File:Tangential_quadrilateral.svg

If a quadrilateral circumscribes a circle, then the sums of its opposite sides are equal.

Above, a + c = b + d.

Rising Above

https://www.google.com/patents/US270295

This is rather unpleasant. In 1883 John Fleck patented a “life-saving apparatus for privy-vaults,” “to provide a means of escape from the vaults should a person by accident fall therein.”

Apparently this was a problem. “In various sections of the United States deep vaults are commonly used, generally constructed of masonry, and, as is often the case, they are but imperfectly covered or otherwise protected at the top to guard against persons falling therein.”

Fleck’s solution is basically a ladder set into the wall of the vault. Good for him, I guess. I hope he wasn’t inspired by some personal tragedy.

A Distilled Impression

kettle anamorphosis

Henry Kettle painted this pyramid anamorphosis around 1770. If a mirrored pyramid is placed at the center of the canvas, then each of its sides reflects a portion of one of the four distorted heads … producing a true hidden portrait when viewed from above.

Big Time

Some notable clock faces: the Metropolitan Life Insurance Company Tower in Manhattan (upper left), the Palace of Westminster in London (upper right), the Allen-Bradley Clock Tower in Milwaukee (lower left), the Spasskaya Tower at the Kremlin (lower right), and the Abraj Al Bait tower in Mecca (center).

Unbelievably, these are shown to scale. Each of the four faces on the Abraj Al Bait is 43 meters square; the minute hand alone is 22 meters long.

The Palace of Westminster is unusual in that its clock uses the numeral IV — most clocks with Roman numerals use IIII in the fourth position, for unclear reasons.

The Traveler’s Dilemma

Two travelers are transporting identical antiques. Unfortunately, the airline smashes both of them. The airline manager proposes that each traveler write down the cost of his antique, any value from $2 to $100. If both write the same number, the airline will pay this amount to both travelers. If they write different numbers, the airline will assume that the lower number is the accurate price; the low bidder will receive this amount plus $2, and the high bidder will receive this amount minus $2. If they can’t confer, what strategy should the travelers take in deciding how to bid?

At first Traveler A might like to bid $100, the maximum allowed. If his opponent does the same then they’ll both net $100. But A can do better than this: If B bids $100 and A bids $99 then A will come away with $101.

Unfortunately if B is rational then he’ll have the same insight and also bid $99. So A had better undercut him again and bid $98.

This chain leads all the way down to $2. If both travelers are perfectly rational then they’ll both bid (and make) $2, the minimum price.

But this seems very unlikely to happen in actual practice — in real life both travelers would likely make high bids and get high (though perhaps unequal) payoffs.

“All intuition seems to militate against all formal reasoning in the traveler’s dilemma,” wrote economist Kaushik Basu in propounding the problem in 1994. “There is something very rational about rejecting (2, 2) and expecting your opponent to do the same. … The aim is to explain why, despite rationality being common knowledge, players would reject (2, 2), as intuitively seems to be the case.”

Unquote

“I am by heritage a Jew, by citizenship a Swiss, and by makeup a human being, and only a human being, without any special attachment to any state or national entity whatsoever.” — Albert Einstein

“I want to be a human being, nothing more and nothing less. … I don’t suppose we can ever stop hating each other, but why encourage that by keeping the old labels with their ready-made history of millennial hate?” — Isaac Asimov

“Patriots always talk of dying for their country, and never of killing for their country.” — Bertrand Russell

“If I knew of something that could serve my nation but would ruin another, I would not propose it to my prince, for I am first a man and only then a Frenchman … because I am necessarily a man, and only accidentally am I French.” — Montesquieu

“Patriotism is your conviction that this country is superior to all other countries because you were born in it.” — George Bernard Shaw

A Hidden Drama

http://en.wikipedia.org/wiki/File:98_rock,_Wake_Island.jpg

When Japanese forces took Wake Island in December 1941, they put 98 captured Americans to forced labor. When the American military threatened to retake the island in October 1943, the Japanese took these civilians to the north end of the island, blindfolded them, and executed them by machine gun.

One of the Americans somehow escaped this massacre and fled. We know this because he returned to the scene and carved the legend “98 US PW 5-10-43″ on a coral rock near the mass grave. He was subsequently captured and beheaded.

The rock remains a landmark on the island. The American’s identity has never been discovered.

A Blind Pirate

http://commons.wikimedia.org/wiki/File:TI-parrot.jpg

Several years after publishing Treasure Island, Robert Louis Stevenson was abashed to discover that he had drawn much of the story from Washington Irving’s 1824 book Tales of a Traveller, which he had read many years earlier and forgotten.

“I believe plagiarism was rarely carried farther,” he wrote later. “The book flew up and struck me: Billy Bones, his chest, the company in the parlour, the whole inner spirit, and a good deal of the material detail of my first chapters — all were there, all were the property of Washington Irving. But I had no guess of it then as I sat writing by the fireside, in what seemed the spring-tides of a somewhat pedestrian inspiration; nor yet day by day, after lunch, as I read aloud my morning’s work to the family.”

This is an instance of cryptomnesia, the mistaking of a forgotten memory for an original idea. Stevenson charged himself with plagiarism, but he had honestly believed he was writing a new story: “It seemed to me original as sin; it seemed to belong to me like my right eye.” In reading Nietzsche’s Thus Spake Zarathustra, Carl Jung was surprised to discover “almost word for word” an incident reported in a ship’s log in 1686. Jung recognized the passage from a book published around 1835, about 50 years before Nietzsche was writing. He contacted the philosopher’s sister, who confirmed that the two of them had read the book when Nietzsche was 11 years old.

“I think, from the context, it is inconceivable that Nietzsche had any idea that he was plagiarizing this story,” Jung wrote. “I believe that fifty years later it had unexpectedly slipped into focus in his conscious mind.”

Orderly

http://commons.wikimedia.org/wiki/File:Incircles.JPG

A Japanese geometry theorem from the Edo period: If the blue circles are equal, the green circles will be equal too.

This can be extended: Circles spanning three of these triangles will also be equal, and so on.

You Rang?

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

Write out the phrase “expect the devil.”

Extract the Roman numerals: eXpeCt the DeVIL

Add these: D (500) + C (100) + L (50) + X (10) + V (5) + I (1)

The total is 666.

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