Visual Calculus

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As a circle rolls along a line, a point on its circumference traces an arch called a cycloid. The arch encloses an area three times that of the circle, a result commonly proven using calculus. Now Armenian mathematician Mamikon Mnatsakanian has devised a “sweeping-tangent theorem” that accomplishes the same proof using intuition:

Imagine a tangent to the rolling circle. As the circle rolls, the tangent sweeps out a series of vectors (approximated here using colors). If these vectors are then gathered to a common point while preserving their length and orientation, they form a sort of bouquet whose size and shape turn out to match exactly those of the original circle. Because the enclosing rectangle has four times the area of the rolling circle (2πr × 2r = 4πr2), this shows that the area under the arch has three times the circle’s area.

All this is proven rigorously in Mnatsakanian’s 2012 book New Horizons in Geometry, written with his Caltech colleague Tom Apostol. The two have now collaborated on some 30 papers showing that many surprising and useful results that heretofore had required integration can now be obtained using intuitive methods that can appeal even to a young student.

That’s a welcome outcome for Mnatsakanian, who found himself stranded in the United States when the Armenian government collapsed in 1990. Apostol writes, “When young Mamikon showed his method to Soviet mathematicians they dismissed it out of hand and said ‘It can’t be right. You can’t solve calculus problems that easily.'”

Constellation

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Circling the earth aboard Friendship 7 in 1962, John Glenn had an odd encounter:

The strangest sight of all came with the very first ray of sunrise as I was crossing the Pacific toward the U.S. I was checking the instrument panel and when I looked back out the window I thought for a minute that I must have tumbled upside-down and was looking up at a new field of stars. I checked my instruments to make sure I was right-side-up. Then I looked again. There, spread as far as I could see, were literally thousands of tiny luminous objects that glowed in the black sky like fireflies. I was riding slowly through them, and the sensation was like walking backwards through a pasture where someone had waved a wand and made all the fireflies stop right where they were and glow steadily. They were greenish yellow in color, and they appeared to be about six to 10 feet apart. I seemed to be passing through them at a speed of from three to five miles an hour. They were all around me, and those nearest the capsule would occasionally move across the window as if I had slightly interrupted their flow. On the next pass I turned the capsule around so that I was looking right into the flow, and though I could see far fewer of them in the light of the rising sun, they were still there. Watching them come toward me, I felt certain they were not caused by anything emanating from the capsule. I thought perhaps I’d stumbled into the lost batch of needles the Air Force had tried to set up in orbit for communications purposes. But I could think of no reason why needles should glow like fireflies, nor did they look like needles. As far as I know, the true identity of these particles is still a mystery.

They seem to have been ice crystals, flakes of frost shed by the capsule and illuminated by the sun. Scott Carpenter saw a similar display on the next Mercury mission, Aurora 7, a few months later.

(John Glenn, “If You’re Shook Up, You Shouldn’t Be There,” Life, March 9, 1962.)

The Peace Arch

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The border between the United States and Canada blurs a bit between Blaine, Wash., and Surrey, B.C. There stands the Peace Arch, a 20-meter monument to amity between the two nations commissioned by railroad executive Sam Hill in 1921.

The arch stands precisely on the border, at the center of an international park: Citizens of either nation can pass without passport or visa into the other nation’s territory, provided they don’t stray beyond a dedicated area.

The U.S. side of the arch bears the inscription “Children of a common mother,” and the Canadian side reads “Brethren dwelling together in unity.” An iron gate stands open on either side, and an inscription above reads “May these gates never be closed.”

Two Futility Closet Books!

FC book covers

Futility Closet: An Idler’s Miscellany of Compendious Amusements and Futility Closet 2: A Second Trove of Intriguing Tidbits, are available now at Amazon. Both contain hundreds of hand-picked favorites from our 10-year archive of curiosities — perfect for your winter solstice peculiar-fact-sharing needs. Some sample items from the first book’s index:

gaiety, of bespectacled horses, 35
hallucination, useful in enlivening Bristol, 94
Nazis, poorly informed regarding Irish railway schedules, 36
chili sauce, and estate planning, 25
softball, at North Pole, 67
whinging, vaudevillian, placated, 86
corned beef sandwiches, weightless, 170
Nagel, Conrad, induced to reflect wearily upon his romantic choices, 116

Boing Boing founder Mark Frauenfelder says, “Futility Closet delivers concentrated doses of weird, wonderful, brain-stimulating ideas and anecdotes, curated mainly from forgotten old books. I’m hooked — there’s nothing quite like it!”

Hesiod’s Anvil

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How far off is heaven? In the Theogony Hesiod gives us a clue:

For a brazen anvil falling down from heaven nine nights and days would reach the earth upon the tenth; and again, a brazen anvil falling from earth nine nights and days would reach Tartarus upon the tenth.

How far can an anvil fall in nine days? Galileo, who taught that “the distances measured by the falling body increase according to the squares of the time,” would have determined that the anvil starts 2.96 × 109 km from earth, a distance greater than that between the sun and Uranus.

But Galileo’s calculation assumes that gravitational force is independent of the object’s distance from the earth. If we assume instead that it varies inversely with the square of the distance between mass centers (and if we ignore all masses except those of the earth and the anvil, and assume that the anvil falls in a straight line), King College mathematician Andrew Simoson calculates that Galileo’s anvil wouldn’t reach us for

hesiod's anvil calculation

Instead, under this new assumption, to reach us in nine days an anvil would start 5.81 × 105 km away — about one and a half times the distance between the earth and the moon.

(Andrew J. Simoson, Hesiod’s Anvil, 2007.)

Reflection

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“Sir Winston Churchill once told me of a reply made by the Duke of Wellington, in his last years, when a friend asked him: ‘If you had your life over again, is there any way in which you could have done better?’ The old Duke replied: ‘Yes, I should have given more praise.'” — Bernard Montgomery, A History of Warfare, 1968

Noted

Further excerpts from the notebooks of English belletrist Geoffrey Madan (1895-1947):

Two psychiatrists meeting: “You’re pretty well, how am I?”

Children: unable to understand the concept of uncertainty.

1. Every subject of the Crown is entitled to make pickles.
2. Every man must bear the name of his father.

— Sir John Markham, Chief Justice, 1465

Pedantry is greater accuracy than the case requires.

“Drink … prevents you seeing yourself as others see you.” — Desmond MacCarthy

Safe remarks:

1. To inaudible remark: “That’s just what I’ve been wondering all the evening.”
2. “I can never remember how you spell your name.” (But G.M. Young quoted the man who wearily replied, “Still J-O-N-E-S.”)

“So they went forth both, and the young man’s dog with them.” — Tobit 5:16: the only mention in the Bible of a pet animal

The dust of exploded beliefs may make a fine sunset.

See Observations and More Madan.

Full Circle

For 88 years the Memorial Bridge carried traffic across the Piscataqua River between Portsmouth, N.H., and Kittery, Maine.

At its opening in 1923, 5-year-old Eileen Foley cut the ribbon.

In 2011, Foley, now 93, tied a ribbon at the closing ceremony.

In the interval she had served several terms as mayor of Portsmouth. “Thank you very much for this afternoon,” she said. “I will never forget it.”

(Thanks, Zach.)

Podcast Episode 34: Spring-Heeled Jack — A Victorian Supervillain

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Between 1837 and 1904, rumors spread of a strange bounding devil who haunted southern England, breathing blue flames and menacing his victims with steel talons. In the latest Futility Closet podcast we review the career of Spring-Heeled Jack and speculate about his origins.

We also recount Alexander Graham Bell’s efforts to help the wounded James Garfield before his doctors’ treatments could kill him and puzzle over why a police manual gives instructions in a language that none of the officers speak.

Source for our segment on Spring-Heeled Jack:

Mike Dash, “Spring-Heeled Jack: To Victorian Bugaboo From Suburban Ghost,” Fortean Studies 3 (1996).

Sources for our segment on Alexander Graham Bell and James Garfield:

Candice Millard, Destiny of the Republic: A Tale of Madness, Medicine and the Murder of a President, 2011.

Amanda Schaffer, “A President Felled by an Assassin and 1880’s Medical Care,” New York Times, July 25, 2006.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

In a Word

2014-11-17-in-a-word

brumal
adj. wintry

hibernal
adj. of, pertaining to, or proper to winter

hiemal
adj. of or belonging to winter

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