Paper Work

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

Rutgers mathematician E.P. Starke posed this question in the American Mathematical Monthly of July 1940:

“In high school geometry texts and elsewhere one frequently meets the statement that the reason for the straightness of the crease in a folded piece of paper is that the intersection of two planes is a straight line. This is fallacious. What is the correct reason?”

I was going to post this as a puzzle, but after much pondering I’ve been unable to make sense of the answer. Here it is:

“Let P, P′ be two points of the paper that are brought into coincidence by the process of folding. Then any point A of the crease is equidistant from P, P′, since the lines AP, AP′ are pressed into coincidence. Hence the crease, being the locus of such points A, is the perpendicular bisector of PP′.”

I agree that this is true, but I don’t see what’s wrong with the first answer. Any ideas?

UPDATE: The consensus seems to be that the first answer makes some invalid assumptions, including flat planes and Euclidean space, where Starke’s proof is more rigorous. Thanks to everyone who’s written in.

(Second update, on reflection: Presumably the books that Starke mentions were not claiming that all creases must be straight, only that a straight crease is so because two planes intersect in a line. That still seems reasonable to me.)

| Science & Math

“A Courtly Spaniard”

While the Duke de Villa Medina was at the English court, he was present, and took part at a tournament given by Elizabeth, where his gallantry and manly beauty made him the observed of all observers. At the close of the sports, as the duke came near to the queen, she said to him, pleasantly, that she would like to know who was the chosen mistress of so gallant a knight; whereupon he shook his head and would not further answer.

‘But,’ persisted Elizabeth, ‘there must be, somewhere, a lady whose beauty and perfection of character gives to her a deeper place in your heart than is yielded to another?’

‘Ah! yes gracious madam; there is one such.’

‘And may I know who she is?’

The duke reflected a moment, and then answered that he would inform her on the morrow.

And on the morrow he sent to the queen inclosed in a box of sandal-wood and mother-of-pearl a small mirror.

Those who know Elizabeth’s character can well imagine how deeply this exquisite bit of flattery must have touched her.

— The Lamp, 1881

| Society

Intercepted

In chess, a pawn may be captured “in passing” — when a pawn advances two squares from its initial position, it may be captured by an adjacent pawn as if it had advanced only one square.

This can lead to a curious state of affairs:

fraenkel en passant chess problem

From this position White plays 1. Bg2+ and declares checkmate. Black says “Au contraire,” plays 1. … d5, and announces checkmate himself. White shakes his head, plays 2. cxd6 e.p., and reasserts his own claim:

fraenkel en passant chess problem

Black claims that this last move is absurd. He says the game ended when he advanced his pawn to d5. But White argues that the pawn never reached d5 — in principle it was captured on d6, and thus could not stop White’s original mate.

So who won the game? It would seem to be a matter of opinion!

From Heinrich Fraenkel, Adventure in Chess, 1951.

| Oddities

An Expansive Idea

http://www.google.com/patents/about?id=ajRFAAAAEBAJ

While serving in Congress in 1848, Abe Lincoln conceived a way to help boats that became stranded on sandbars. If bellows were attached to a craft below the waterline, these could be inflated when it got stuck, buoying the craft and allowing it to float over the shoal.

Lincoln whittled a 20-inch model from a cigar box and a shingle. His law partner, W.H. Herndon, didn’t think much of it, but Lincoln presented it to lawyer Z.C. Robbins, who arranged a patent in 1849. This makes Lincoln the only president to hold a patent.

Apparently it never went to market, though. “Railroads soon diverted traffic from the rivers,” Robbins recalled, “and Lincoln got deep in law and politics, and I don’t think he ever received a dollar from it.”

| History · Technology

Lost and Found

http://commons.wikimedia.org/wiki/File:Charles_Dickens_-_Project_Gutenberg_eText_13103.jpg

Charles Dickens, during one of his visits to Paris, had his watch stolen from him at the theatre. This watch had been given to him by the Queen, and was, therefore, very much prized by him. On returning to his hotel, Dickens found a small parcel waiting him, to which was pinned the following note:–

Sir,–I hope you will excuse me, but I assure you I thought I was dealing with a Frenchman and not a countryman. Finding out my mistake, I hasten to repair it as much as lies in my power, by returning you herewith the watch I stole from you. I beg you to accept the homage of my respect, and to believe me, my dear countryman, your humble and obedient servant,

A PICKPOCKET.

— The Dickensian, September 1906

| Crime · Literature

Small Press

The first eyewitness account of the Wright brothers’ flying machine appeared in the journal Gleanings in Bee Culture.

The editor, beekeeper Amos I. Root, had visited the Wrights in 1904 at Huffman Prairie, Ohio, where they were working to perfect the machine after its historic first flight the preceding December.

Root sent copies of his article to Scientific American — but they were dismissed.

| History · Oddities · Technology

Stamps of Character

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

Locked between India and Tibet, the tiny kingdom of Bhutan has a curious claim to distinction: its postage stamps.

In 1951 American entrepreneur Burt Todd became one of the first Westerners to visit the Himalayan nation, and he devised the stamp program explicitly to help expand the country’s economic base.

There followed two decades of increasingly bizarre postage: 3-D stamps; stamps scented like roses; stamps with textured brushstrokes and bas-reliefs; stamps printed on stainless steel, silk, and extruded plastic; even “talking stamps,” discs of grooved rubber that can be played on a phonograph (one plays the national anthem, another contains a fleeting spoken history of Bhutan).

Todd lost his contract in 1974, and the country moved into more conventional postage. But the tradition isn’t entirely over: In 2008, Todd’s daughter arranged the world’s first CD-ROM postage stamp — it plays a video recounting the history of Bhutanese kings.

| Oddities

The Consensus

Mrs. H.A. Deming spent a year assembling lines from 38 English and American poets into this mosaic verse, published originally in the San Francisco Times in the 19th century:

“Life”

Why all this toil for triumphs of an hour? [Young]
Life’s a short summer–man is but a flower. [Dr. Johnson]
By turns we catch the fatal breath and die; [Pope]
The cradle and the tomb, alas! how nigh. [Prior]
To be better far than not to be, [Sewell]
Though all man’s life may seem a tragedy; [Spencer]
But light cares speak when mighty griefs are dumb– [Daniel]
The bottom is but shallow whence they come. [Sir Walter Raleigh]
Thy fate is the common fate of all; [Longfellow]
Unmingled joys here no man befall; [Southwell]
Nature to each allots his proper sphere, [Congreve]
Fortune makes folly her peculiar care. [Churchill]
Custom does often reason overrule, [Rochester]
And throw a cruel sunshine on a fool. [Armstrong]
Live well; how long or short permit to Heaven. [Milton]
They who forgive most shall be most forgiven. [Bailey]
Sin may be clasped so close we cannot see its face– [French]
Vile intercourse where virtue has no place; [Somerville]
Then keep each passion down, however dear, [Thompson]
Thou pendulum betwixt a smile and tear. [Byron]
Her sensual snares let faithless pleasure lay, [Smollett]
With craft and skill to ruin and betray; [Crabbe]
Soar not too high to fall, but stoop to rise; [Massinger]
We masters grow of all that we despise. [Crowley]
Oh, then, renounce that impious self-esteem. [Beattie]
Riches have wings and grandeur is a dream. [Cowper]
Think not ambition wise because ’tis brave, [Sir William Davenant]
The paths of glory lead but to the grave; [Gray]
What is ambition? ‘Tis a glorious cheat, [Wills]
Only destructive to the brave and great. [Addison]
What’s all the gaudy glitter of a crown? [Dryden]
The way to bliss lies not on beds of down. [Francis Quarles]
How long we live, not years, but actions tell; [Watkins]
That man lives twice who lives the first life well. [Herrick]
Make, then, while yet ye may, your God your friend, [William Mason]
Whom Christians worship, yet not comprehend. [Hill]
The trust that’s given guard, and to yourself be just, [Dana]
For live we how we may, yet die we must. [Shakespeare]

| Literature · Poems

Charmed

http://commons.wikimedia.org/wiki/File:Niels_Bohr_1935.jpg

A visitor to Niels Bohr’s cottage noticed a horseshoe nailed over the door.

“Surely you don’t expect that a horseshoe will bring good luck?” asked the visitor.

“No, I don’t,” Bohr said. “But they say it works even if you don’t believe in it.”

See The Misfortune Field.

| Science & Math