All Covered

Can any 10 points on a plane always be covered with some number of nonoverlapping unit discs?

It’s not immediately clear that the answer is yes. The most efficient way to pack circles together on the plane is the hexagonal packing shown above; it covers about 90.69 percent of the surface. But if our 10 points are inconveniently arranged, it’s not clear that we’ll always be able to shift the array of circles around in order to get them all covered.

In this case there’s a neat proof that takes advantage of a technique called the probabilistic method — if, for a group of objects, the probability is less than 1 that a randomly chosen object does not have a certain property, then there must exist an object that has this property.

Take a hexagonal packing randomly. Then, for any point on the plane, the probability that it’s not covered by the chosen packing is about 1 – 0.9069 = 0.0931. This means that for any 10 points, the chance that one or more points are not covered is approximately 0.0931 × 10 = 0.931. And that’s less than 1.

“Therefore, we obtain from the principle that there exists some closest packing that covers all the 10 points,” writes mathematician Hirokazu Iwasawa of the Institute of Actuaries of Japan. “And, in such a packing, we actually need at most 10 discs to cover the 10 points.”

(Hirokazu Iwasawa, “Using Probability to Prove Existence,” Mathematical Intelligencer 34:3 [September 2012], 11-14. The puzzle was created by Naoki Inaba.)

May 13, 2022May 15, 2022 | Science & Math

Gödel’s Loophole

At Princeton in the 1940s, Albert Einstein became a close friend of logician Kurt Gödel, whose incompleteness theorems lie at the heart of modern mathematics. Toward the end of his life Einstein said that his “own work no longer meant much, that he came to the Institute merely … to have the privilege of walking home with Gödel.”

In 1947 Einstein and economist Oskar Morgenstern accompanied Gödel to his U.S. citizenship exam because they were concerned about his unpredictable behavior: During his voluminous preparation for the exam, Gödel said, he had uncovered a flaw in the U.S. constitution that could lead to a dictatorship. Einstein and Morgenstern told him that the exam would really be quite simple and urged him not to prepare so extensively.

At the hearing, judge Phillip Forman asked Gödel:

“Now, Mr. Gödel, where do you come from?”

“Where I come from? Austria.”

“What kind of government did you have in Austria?”

“It was a republic, but the constitution was such that it finally was changed into a dictatorship.”

“Oh! That is very bad. This could not happen in this country.”

“Oh, yes,” Gödel said. “I can prove it.”

“So of all the possible questions, just that critical one was asked by the Examinor,” Morgenstern wrote later. “Einstein and I were horrified during this exchange; the Examinor was intelligent enough to quickly quieten Gödel and say, ‘Oh, God, let’s not go into this.'”

The logician got his citizenship and the friends returned to Princeton. What was the flaw that Gödel had found? There’s no record of it in Morgenstern’s account, so we don’t know. Stephen Hawking suggests that it involved the president’s power to fill vacancies during Senate recesses, and Barry University law professor F.E. Guerra-Pujol conjectures that it might involve the constitution’s power to amend itself. Maybe it’s best if we never discover it.

(Thanks, Louis.)

May 12, 2022May 11, 2022 | History · Science & Math · Society

Round Numbers

I found this surprising. What’s the volume of a ball of radius 1 in various dimensions?

In one dimension it’s a line segment of length 2.

In two dimensions it’s a unit disc in the plane, with area π.

In three dimensions it’s a unit ball with volume 4π/3.

Intuitively we might expect the number to keep rising. But it doesn’t!

In fact it peaks at five dimensions, and it drops quite sharply after that. In 20 dimensions the volume is only 0.026, and the limiting value is zero. Wikipedia explains the math.

May 11, 2022May 11, 2022 | Science & Math

Deep Thinking

https://commons.wikimedia.org/wiki/File:Florence_-_Graffiti_-_Mona_Lisa_Underwater.JPG — Image: Wikimedia Commons

Natural philosopher John Wilkins’ Mathematical Magick of 1648 contains a startling passage in which he foretells the advantages of a long-range submarine, or “ship, wherein men may safely swim underwater”:

‘Tis private; a man may thus go to any coast of the world invisibly, without being discovered or prevented in his journey;
‘Tis safe; from the uncertainty of Tides, and the violence of Tempests, which do never move the sea above five or six paces deep. From Pirates and Robbers which do so infest other voyages; from ice and great frosts, which do so much endanger the passages toward the Poles.
It may be of very great advantage against a Navy of enemies, who by this means may be undermined in the water, and blown up.
It may be of a special use for the relief of any place that is besieged by water, to convey unto them invisible supplies: and so likewise for the surprisal of any place that is accessible by water.
It may be of unspeakable benefit from submarine experiments and discoveries.

Wilkins was aware of Cornelius Drebbel’s primitive sub of 1620, but he looks much farther ahead, seeming to foresee combat submarines and deep-sea exploration vessels.

“I am not able to judge what other advantages there may be suggested, or whether experiment would fully answer to these notional conjectures,” he concluded. “But however, because the invention did unto me seem ingenious and new, being not impertinent to the present enquiry, therefore I thought it might be worth the mentioning.”

(From Joseph J. Thorndike Jr., ed., Mysteries of the Deep, 1980.)

May 11, 2022May 10, 2022 | History · Technology

Left or Right?

A curious physics puzzle from Mark Levi’s excellent Why Cats Land on Their Feet: Suppose two astronauts, Al and Bob, are strapped to opposite ends of a space capsule’s interior, Al on the left and Bob on the right. Al is holding a large helium balloon, and everything is at rest. If Al pushes the balloon toward Bob, which way will the capsule drift?

It would be reasonable to guess that the capsule will drift to the left. Newton’s third law says that action equals reaction, so as Al pushes the balloon to the right, the balloon pushes Al to the left, and since he’s strapped to the capsule, he and it should drift left.

In fact the capsule will drift right as well. Because there are no external forces, the center of mass of the whole system is fixed. The helium balloon has less mass than the air it displaces, so from Al’s point of view the center of mass moves left. But the center of mass of the whole system is fixed in space, so the capsule must move right from the point of view of an external observer.

One way to make this intuitive is to imagine that the capsule is full of water rather than air. The mass of water essentially stays in place while we transfer a bubble of helium from the water’s left to its right. To accommodate this, the shell (whose mass we neglect) must move to the right.

May 10, 2022May 9, 2022 | Science & Math

Face to Face

Japanese pilot Kaname Harada recalls air combat in World War II:

The initial feeling after shooting down someone was relief, because it was not me who was shot down. My next thought was that I was a better pilot, so I felt superior to the enemy aviator who was shot down. These feelings lasted only for a short time. When we shot at each other, we were at very close range, and during this time I could see my opponent’s face very well. When I saw the enemy’s face, it looked terrible because he was going down. Soon after this I felt very bad, because I could imagine that my opponent had a family of his own, and I killed him. Therefore, to this day I feel very bad about shooting down pilots during the war.

Afterward, when these faces haunted his dreams, Harada became an antiwar activist and even traveled to the United States and Britain to meet some of the pilots he’d fought against. “In general, I have a bad feeling about being involved in the war, and I feel guilty about killing other people in combat,” he said. “I also feel that the war should not have happened in the first place. This is because the governments of countries around the world don’t make an effort to resolve their differences. Instead, they order their armed forces to kill each other. I believe World War II veterans know this best, because we were used as ‘pawns’ by our government to fight a war.”

(From Ron Werneth, Beyond Pearl Harbor, 2008.)

May 10, 2022May 9, 2022 | History

Missive

In May 1936 a publisher invited Albert Einstein to contribute a message to be sealed in a metal box in the cornerstone of a new library wing in his country home, to be opened a thousand years hence. He sent this:

Dear Posterity,

If you have not become more just, more peaceful, and generally more rational than we are (or were) — why then, the Devil take you.

(From Helen Dukas and Banesh Hoffmann, eds., Albert Einstein, the Human Side, 1979.)

May 9, 2022May 8, 2022 | History · Science & Math

Glass Town

In June 1829, English curate Patrick Brontë brought home a box of 12 wooden toy soldiers for his 12-year-old son Branwell. Branwell shared them with his sisters: “This is the Duke of Wellington! It shall be mine!” shouted 13-year-old Charlotte, and 11-year-old Emily and 9-year-old Anne soon took up avatars of their own. This was the start of an enormous imaginative undertaking — soon the four had invented names and personalities for their soldiers and had begun inventing a shared history in which the “Young Men” traveled to the west coast of Africa; settled there after a war with the indigenous Ashantee; elected Arthur Wellesley, the Duke of Wellington, as their leader; and founded the Great Glass Town at the delta of the River Niger.

This was just the beginning. After 1831 Emily and Ann “seceded” to create a related imaginary country they called Gondal, and after 1834 Charlotte and Branwell developed Glass Town into Angria, yet another imaginary nation. Together and variously they edited magazines, wrote histories, and composed stories, poems, and plays about these shared fantasy world, with alliances, feuds, and love affairs that play out across Africa and the Pacific. Here’s the start of “A Day at Parry’s Place,” written by 14-year-old Charlotte in a fanciful magazine in 1830:

‘Oh, Arthur!’ said I, one morning last May. ‘How dull this Glass Town is! I am positively dying of ennui. Can you suggest anything likely to relieve my disconsolate situation?’

‘Indeed, Charles, I should think you might find some pleasant employment in reading or conversing with those that are wiser than yourself. Surely you are not to emty-headed & brainless as to be driven to the extremity of not knowing what to do!’ Such was the reply to my civil question, uttered with the prettiest air of gravity imaginable.

‘Oh, yes! I am, brother! So you must furnish me with some amusement.’

‘Well, then, Charles, you have often spoken of a visit to Captain Parry’s Palace as a thing to be desired. You have now time for the accomplishment of your wish.’

Together, four young siblings in a quiet parsonage in Haworth filled 484 pages documenting their imaginary world before maturity sent them on to other pursuits. “As the sisters grew older, Anne — once as close as a twin — gradually ceased to share Emily’s personal vision of the saga, just as the partnership between Charlotte and Branwell slowly disintegrated as their interests and aesthetic vision changed with maturity,” writes Christine Alexander in Tales of Glass Town, Angria, and Gondal (2010). “[But] the young Brontës nourished each other’s imaginations and developed in their youthful writings the independent styles and themes that can be seen fully developed in their mature poetry and famous novels.”

May 9, 2022May 8, 2022 | Literature

Self-Regard

Aristotle referred to humans as the “state-building animal.” Other names our species has proposed for itself, in various writings:

Homo absconditus: “man the inscrutable”
Homo adorans: “worshipping man”
Homo aestheticus: “aesthetic man”
Homo amans: “loving man”
Homo animalis: “man with a soul”
Homo avarus: “man the greedy”
Homo creator: “creator man”
Homo demens: “mad man” (the only creature with irrational delusions)
Homo discens: “learning man”
Homo domesticus: “domestic man” (because he builds his environment)
Homo duplex: “double man” (because of his animal and social tendencies)
Homo economicus: “economic man”
Homo educandus: “to be educated” (we require education to reach maturity)
Homo ethicus: “ethical man”
Homo excentricus: “not self-centered” (we’re capable of objectivity and self-reflection)
Homo faber: “toolmaker man”
Homo ferox: “ferocious man” (T.H. White)
Homo grammaticus: “grammatical man”
Homo humanus: “human man” (as opposed to Homo biologicus)
Homo hypocritus: “hypocritical man” (Robin Hanson, who also called us “man the sly rule bender”)
Homo imitans: “imitating man” (capable of learning by imitation)
Homo inermis: “helpless man” (devoid of animal instincts)
Homo investigans: “investigating man” (curious and capable of learning by deduction)
Homo laborans: “working man” (capable of dividing labor and specializing)
Homo logicus: “the man who wants to understand”
Homo loquens: “talking man”
Homo loquax: “chattering man” (Henri Bergson)
Homo ludens: “playing man” (Schiller)
Homo mendax: “lying man” (able to tell lies)
Homo metaphysicus: “metaphysical man” (Schopenhauer)
Pan narrans: “storytelling ape” (Terry Pratchett)
Homo necans: “killing man”
Homo patiens: “suffering man” (Viktor Frankl)
Homo pictor: “depicting man”
Homo poetica: “man the poet”
Homo religiosus: “religious man”
Homo ridens: “laughing man”
Homo reciprocans: “reciprocal man” (a cooperative actor)
Homo sanguinis: “bloody man”
Homo sciens: “knowing man”
Homo sentimentalis: “sentimental man” (empathizing and idealizing emotions)
Homo socius: “social man”
Homo sociologicus: “sociological man” (prone to sociology)
Homo technologicus: “technological man”
Homo viator: “man the pilgrim” (on his way toward finding God)

Wikipedia lists 72 of these. Douglas Adams wrote, “Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so.”

May 8, 2022May 7, 2022 | History · Society

The Rousham Eyecatcher

https://www.geograph.org.uk/photo/77685 — Image: geograph

There’s something beautifully useless about this: The gardens at Rousham House, in Oxfordshire, occupy only 25 acres, so landscape architect William Kent added a sham ruin on the brow of a distant hill to give the impression that the house’s landscape extended far beyond its boundaries.

Horace Walpole wrote that Kent “felt the delicious contrast of hill and valley changing imperceptibly into each other, tasted the beauty of the gentle swell, or concave scoop, and remarked how loose groves crowned an easy eminence with happy ornament, and while they called in the distant view between their graceful stems, removed and extended the perspective by delusive comparison.”

May 7, 2022May 6, 2022 | Oddities