Proof Without Words

schaer proof

In the January 2001 issue of the College Mathematics Journal, University of Calgary mathematician Jonathan Schaer offers this simple proof that arctan 1 + arctan 2 + arctan 3 = π. The sum of the angles of this large triangle is 180°. And the diagram shows that its lower left angle is arctan 3, its lower right angle is arctan 2, and its top angle is part of an isosceles right triangle. So arctan 1 + arctan 2 + arctan 3 = 180°, or π radians. “No words, even no symbols!”

(“Miscellanea,” College Mathematics Journal 32:1, 68-71.)

Rotating Office

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Actress Ilka Chase married Louis Calhern in 1926, but he divorced her the following year to marry Julia Hoyt.

Sorting through her possessions afterward, she discovered a set of engraved calling cards that she’d had printed with the name Mrs. Louis Calhern.

“They were the best cards — thin, flexible parchment, highly embossed — and it seemed a pity to waste them, and so I mailed the box to my successor,” she wrote later.

“But aware of Lou’s mercurial marital habits, I wrote on the top one, ‘Dear Julia, I hope these reach you in time.’

“I received no acknowledgment.”

(From Chase’s 1945 autobiography Past Imperfect.)

In a Word

volant
adj. engaged in flight

espieglerie
n. impish or playful behaviour; mischief

apopemptic
adj. pertaining to leave-taking or departing

réclame
n. publicity or notoriety

In 1953, 61-year-old British ace Christopher Draper flew an Auster monoplane under 15 of the 18 bridges on the Thames, negotiating 50-foot arches at 90 mph.

“I did it for the publicity,” he told the press. “For 14 months I have been out of a job, and I’m broke. I wanted to prove that I am still fit, useful and worth employing. … It was my last-ever flight — I meant it as a spectacular swan song.” He was fined 10 guineas.

Night Life

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Suppose that your dream-life underwent a remarkable change. Suppose that on going to bed at home and falling asleep, you found yourself to all appearances waking up in a hut raised on poles at the edge of a lake. A dusky woman, whom you realize to be your wife, tells you to go out and catch some fish. The dream continues with the apparent length of an ordinary human day, replete with an appropriate and causally coherent variety of tropical incident. At last you climb up the rope ladder to your hut and fall asleep. At once you find yourself awaking at home, to the world of normal responsibilities and expectations. The next night life by the side of the tropical lake continues in a coherent and natural way from the point at which it left off. Your wife says, ‘You were very restless last night. What were you dreaming about?’ and you find yourself giving her a condensed version of your English day. And so it goes on. Injuries given in England leave scars in England, insults given at the lakeside complicate lakeside personal relations. One day in England, after a heavy lunch, you fall asleep in your armchair and dream of yourself, or find yourself, waking up in the middle of the night beside the lake. Things get too much for you at the lakeside, your wife has departed with all the cooking-pots, and you suspect that she is urging the villagers to sacrifice you to the moon. So you fall on your fish-spear and from that moment on your English slumbers are disturbed no more than in the old pre-lakeside days.

— Anthony Quinton, Thoughts and Thinkers, 1982

“Is such a two-space reality conceivable?” asks Peg Tittle in What If…: Collected Thought Experiments in Philosophy (2016). “That is, is it conceivable that we could live in two different but real spaces?”

See Figure and Ground.

Last Words

On Sept. 8, 1880, an explosion tore through the Seaham Colliery, a coal mine in County Durham in the North of England, filling both shafts with debris and trapping scores of men in the burning seams. Most of the 164 dead were found where they had been working, suggesting that death had come quickly, but some of those farther from the shafts appear to have survived for hours or even longer in pockets of air — the oil in some of their lamps was exhausted. Some of these men had had time to leave messages to those who might find them:

September 8 1880
E.Hall and J.Lonsdale died at half-past 3 in the morning.W.Murray and W.Morris and James Clarke visited the rest on half-past nine in the morn and all living in the incline,
Yours truly,
W.Murray, Master-Shifter

Five o’ clock, we have been praying to God

The Lord has been with us, we are all ready for heaven – Ric Cole, half past 2 o’ clock Thursday

Bless the Lord we have had a jolly prayer meeting, every man ready for glory. Praise the Lord. Sign.R.Cole

Dear Margaret,
There was 40 of us altogether at 7am. Some was singing hymns but my thoughts was on my little Michael that him and I would meet in heaven at the same time. Oh Dear wife, God save you and the children, and pray for me … Dear wife, Farewell. My last thoughts are about you and the children. Be sure and learn the children to pray for me. Oh what an awful position we are in ! Michael Smith, 54 Henry Street

The last had been scratched on a tin water bottle with a rusty nail by Michael Smith, who had left his dying infant son to go to work in the mine that morning. The son died on the same day.

(From Durham Records Online.)

Toeholds

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Image: Wikimedia Commons

Set into the corner of Seventh Avenue and Christopher Street in Manhattan’s West Village is a triangular plaque reading PROPERTY OF THE HESS ESTATE WHICH HAS NEVER BEEN DEDICATED FOR PUBLIC PURPOSES. The “Hess triangle” is a remnant from a property dispute that unfolded here 100 years ago: The city was claiming eminent domain in order to demolish hundreds of buildings and expand the subway, but surveyors overlooked this 65-centimeter triangle, owned by Philadelphia landlord David Hess. Hess, outraged at the loss of his five-story apartment building, refused to donate the triangle to the public and added the plaque as a sort of existential revenge. In 1938 it was sold to the adjacent cigar store, and today it’s owned by a local realty corporation, the smallest plot of land in New York City.

Related: In 1973, artist Gordon Matta-Clark bought 13 unused pieces of land that were left over when property lines were redrawn in the borough of Queens. He paid between $25 and $75 for each. The sites are often irregular or isolated, located where other properties meet in a block, some measuring as little as 2×3 feet.

“When I bought those properties at the New York City Auction, the description of them that always excited me the most was ‘inaccessible’,” he said. “They were a group of fifteen micro-panels of land in Queens, leftover properties from an architect’s drawing. One or two of the prize ones were a foot strip down somebody’s driveway and a square foot of sidewalk. And the others were kerbstone and gutterspace. What I basically wanted to do was to designate spaces that wouldn’t be seen and certainly not occupied. Buying them was my own take on the strangeness of existing property demarcation lines. Property is so all-pervasive. Everyone’s notion of ownership is determined by the use factor.”

(Jeffrey Kastner and Brian Wallis, Land & Environmental Art, 2005.)

Who’s Who

John Bevis’ 2010 book Aaaaw to Zzzzzd: The Words of Birds collects the nonsense words that birders have invented to try to convey bird calls and songs:

ag ag ag ag arr: fulmar
beesh: scaled quail
bek bek bek: red-throated loon
chack-weet weet-chack: northern wheater
djadjadja: twite
ee woomp: bittern
hup-hup-a-hwooo: red-billed pigeon
kakakowlp-kowlp: yellow-billed cuckoo
kuk-kuk-cow-cow-cow-cowp-cowp: pied-billed grebe
quickquickquickquick: cuckoo
seedle seedle seedle chup chup: hermit warbler
tiutiu-tiutiutiuk-swee: yellowhammer
trrrrk: wrentit
tzew-zuppity-zuppity-zup: rufous hummingbird
weeta weeta weeta che che che: Lucy’s warbler
wheet-tsack-tsack-tsack: stonechat
zeeda-zeeda-zeeda-sissi-peeso: goldcrest
zoo zee zoo zoo zee: black-throated green warbler

Other interpreters have used actual words — the white-eyed vireo says, “Pick up the beer check quick!”

The Watlington White Mark

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In 1764 Oxfordshire squire Edward Horne decided that the parish church of St Leonard on Watlington Hill would look more impressive with a spire. So he gave it one: By cutting a narrow mark 270 feet long into the chalk soil beyond the building, he was able to perch a foreshortened triangle atop the church when it was viewed in perspective from his house. Local residents maintain the mark to this day.

Blind Leading

“Always let your conscience be your guide,” Jiminy Cricket tells Pinocchio. And so we are constantly telling one another: We seem to believe that following our convictions, whatever they are, is better than “giving in to temptation,” regardless of the outcome. Similarly, resisting temptation and doing what I feel is morally right is somehow praiseworthy, even if it can be shown that my convictions were mistaken. We are saying:

If you believe it is wrong for you to do X then it is wrong for you to do X.

And if you do have such a conviction:

You believe it is wrong for you to do X.

Then combining these statements produces the disconcerting conclusion

It is wrong for you to do X.

This seems to mean that all my convictions about my moral conduct are correct, regardless of the facts of the matter, the substance of my convictions, and even whether I’ve considered them.

“Needless to say, this is deeply implausible,” writes University of New Mexico philosopher G.F. Schueler. “For one thing, not only do these ‘proofs’ not depend on the content of our moral convictions; they don’t depend in any way on how we arrived at these convictions. They ‘prove’ the convictions not only of the moral philosopher who has spent her life seriously reflecting on morality, but also those of the most superficial ditz, who has never read or thought about anything more profound than comic books or video games, not to mention the racist bigot who is convinced that it is wrong for her to allow blacks to vote and the religious zealot who thinks all those who don’t accept her religion should be driven out of the country. They ‘prove’ absolutely everybody’s convictions equally.” And that means we must doubt and study even our own convictions … which is much harder than relying on a cricket.

(G.F. Schueler, “Is It Possible to Follow One’s Conscience?”, American Philosophical Quarterly, 44:1 [January 2007], 51-60.)

A Matter of Degree

A clever conundrum from the U.K. Government Communications Headquarters Christmas Puzzle Quiz — it doesn’t even appear to be a question:

42°15′N 72°15′W, 53°52′N 44°50′E, 37°49′N 85°29′W, 39°37′N 75°56′W, 40°57′N 40°17′E, 51°54′N 02°04′W?

The answer is 52°12′N 1°41′W. The listed coordinates identify Ware, Issa, Bardstown, North-East, Of, and Cheltenham. Where is a Bard’s town northeast of Cheltenham? Stratford-upon-Avon! Its own coordinates make up the answer.

GCHQ is a great fount of excellent original puzzles — some are given on their website, they’ve published two collections, and they regularly post brainteasers on Twitter.