The Sylvester–Gallai Theorem

https://commons.wikimedia.org/wiki/File:Sylvester_gallai_kelly_proof.svg
Image: Wikimedia Commons

Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points.

This proof is by Michigan State University mathematician Leroy Milton Kelly. Consider a set S of points that aren’t all collinear, and define a connecting line to be a line that contains at least two of these points. There must be some point P and connecting line ℓ that are closer together than any other point-line pair in the set. Kelly now proves that ℓ contains only two of the points in S.

Assume that this isn’t true; that is, assume that ℓ contains more than two points in S. Then it passes through at least three points in the set. At least two of these must fall on the same side of P′, the perpendicular projection of P on ℓ. Call these two points B and C, with B being closest to P′. If we draw a connecting line 𝓂 that passes through P and C, and draw the perpendicular from B to B′ on 𝓂, then BB′ will be shorter than PP′ (because PP′C and BB′C are similar triangles).

This is a contradiction — we’d defined P and ℓ as the point-line pair that are closer together than any other pair in the set. So our assumption that ℓ contains more than two points can’t be true.

| Science & Math

The Moral Superiority Illusion

In 2017, University of London psychologists Ben Tappin and Ryan McKay asked 270 subjects to evaluate both themselves and the average person as to traits describing morality (honest, trustworthy, fair), agency (hardworking, knowledgeable, competent), and sociability (cooperative, warm, family-orientated).

They found that nearly all the subjects inflated their own desirable moral qualities irrationally. The more desirable a moral trait appeared to the subjects, the more likely they were to ascribe it to themselves.

“The belief that one is morally superior to the average person appears robust and widespread,” they wrote. “We find that moral superiority represents a uniquely strong and prevalent instance of ‘positive illusion.'”

(Ben M. Tappin and Ryan T. McKay, “The Illusion of Moral Superiority,” Social Psychological and Personality Science 8:6 [August 2017], 623-631.)

| Science & Math · Society

The Tennis Racket Theorem

Experimenting in microgravity aboard the Mir space station in 1985, Soviet cosmonaut Vladimir Dzhanibekov watched a spinning wingnut flip spontaneously in midair. He was observing a phenomenon that had first been described mathematically by Louis Poinsot in 1834 — when a rigid body with three distinct principal moments of inertia is rotated around its intermediate axis, it can exhibit unpredictable 180-degree flips even in the absence of external torques.

This can be demonstrated easily with a tennis racket: When the racket is flipped around its first or third principal axis (the first and third images below), it doesn’t also undergo a half-rotation around another axis. But when it’s flipped around the intermediate axis (center image), it tends to contort itself so that it’s caught with the opposite face upward. (The racket in this demonstration is colored half white and half black to make this clearer.)

https://commons.wikimedia.org/wiki/File:Tennis_racket_theorem.gif
Image: Wikimedia Commons
| Science & Math

Long Distance

de moraes chess puzzle

In this classic puzzle by Florencio Mendes de Moraes, White must checkmate Black in seven moves. The catch is that all three of his queens are confined to the leftmost file.

Either of two first moves will get things started, but beyond that there’s only one sequence that will work. What is it?

Click for Answer
| Puzzles

Out of Joint

https://commons.wikimedia.org/wiki/File:2017-02_Raffaello_Sorbi_-_Time_on_his_hands.jpg

“A man of correct insight among those who are duped and deluded resembles one whose watch is right while all the clocks in the town give the wrong time. He alone knows the correct time, but of what use is this to him? The whole world is guided by the clocks that show the wrong time.” — Schopenhauer, Counsels and Maxims

| Society

A Wide Net

Entries in the index of Charles Cotton’s 1685 translation of Montaigne’s essays:

Ancient History how far respectable
Ants Conference
Arts taught to Men by other Animals
Books, immortal Children
Chearfulness Sign of Wisdom
Chewing accounted unbecoming
Clothes unknown to many Nations
Compositions that smell of Oil and the Lamp
Convenience, none without Inconvenience
Counsel’s Strength consists in Time
Desire of gathering Riches has no Limits
Dowry, great, Ruin of Families
Dying for Fear of Death
Elephant, discovering the Cheat of his Keeper
Feet performing the Service of Hands
Fortune often coincides with Reason
Foxes judging of the Thickness of the Ice by hearing
Goats trained up to give suck to Children
Greyhound’s Imagination
Head uncovered in the Presence of the Gods, why
Hedgehogs Foreknowledge of the Wind
Hospitals for Beasts
Kings and Coblers Souls cast in the same Mold
Lice cracked with the Teeth
Life, many Ways to get rid of it
Magpie that imitated the Trumpet
Monsters, whether there are such properly
Nations that sleep and wake by Half Years
Nobility, what
Opinion gives Weight and Value to Things
Parchment Scrolls tied about the Neck of Pericles
Parents dead Bodies eaten by their Children
Pedants despised
Pleasures wheedle and caress only to strangle us
Poverty obstinate
Prince ought to be himself in the Field
Revenge against inanimate Creatures
Skin of a Man, sufficient Proof against Weather
Trees buried in Winter
Utility, public, too dearly purchased
Women are more learned than Men in Love’s Business
Word of Men, their only Tye on one another
World a Looking-Glass and Book

The full index (in three volumes) is here, here, and here. It was added after Cotton’s death; I’m not sure who composed it.

| Literature

Quinary

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When Saint John of Nepomuk was martyred, legend claims that five stars appeared above his head. So in 1721, when Czech architect Jan Santini Aichel designed a church to commemorate the Bohemian clergyman, he gave it a fivefold symmetry: The circular nave is surrounded by five pairs of columns and five oval domes alternating with ogival apses. The primary altar (one of five) bears five angels supporting a heavenly sphere decorated with five stars.

It’s beautiful even from above — the surrounding ring cloister is divided into 10 sections by five chapels and five gates:

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Image: Wikimedia Commons
| Art · Religion · Science & Math

Science Marches On

From a 1915 examination of Darkie, a blind New Zealand terrier who barked answers to his master’s queries:

https://archive.org/details/journalofsociety17soci/page/100/mode/2up

From the local Lyttelton Times, June 12:

https://archive.org/details/journalofsociety17soci/page/104/mode/2up

Apparently the dog was also clairvoyant:

https://archive.org/details/journalofsociety17soci/page/104/mode/2up

Further questions successfully answered: “How many roods in one acre? How many furlongs in one mile? How many ounces in one pound avoirdupois? How many troy ounces? How many lbs. in one stone? What is the day of the month? What is the month of the year? What is the year over 1900?”

The Society for Psychical Research raised the prospect that the dog was merely continuing to respond until it received a cue to stop, as other performing animals had been discovered to do. It asked its New Zealand correspondent to test this hypothesis, but as far as I can tell he never did.

| Oddities