“Mr. Chairman, before we put the motion ‘That the motion be now put,’ should we not first put the motion ‘That the motion “That the motion be now put” be now put’?” — Punch, 1952
Science & Math
Polchinski’s Paradox
In 1990, physicist Joseph Polchinski described a puzzling situation: Suppose a billiard ball were sent through a traversable wormhole into the past, where it knocked its earlier self off course, away from the wormhole. This is a version of the famous grandfather paradox: How can we explain the ball’s presence where it seems to have prevented its own arrival?
Fortunately, Caltech students Fernando Echeverria and Gunnar Klinkhammer found that self-consistent solutions exist in which the older ball imparts only a glancing blow to its younger self, one that sends it through the hole at just the right angle to deliver a similarly glancing blow to its own forebear, and so on, sparing us from the paradox. (In 1991 the students and Kip Thorne reported that they’d failed to find any initial conditions in which such self-consistent solutions were unavailable, so it’s plausible that they exist for every possible initial trajectory. But this hasn’t been proven.)
Another pleasing possibility is that the older ball knocks the younger out of the way and itself enters the wormhole again. This creates an infinite loop, and some existential bewilderment, but no paradox.
Friendly Numbers
The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Add those up and divide the sum by 30 itself and you get 12/5.
The divisors of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Add those up and divide the sum by 140 and you also get 12/5.
This makes 30 and 140 a friendly pair. Other members of their little club are 2480, 6200, and 40640.
Interestingly, no one knows whether 10 has such a partner. If it does, its smallest friend is at least 1030.
Faith and Hope
In 1915, British evangelist Elizabeth Reid Cotton claimed that she had visited Charles Darwin shortly before his death in 1882, and that he’d told her he regretted advancing the theory of natural selection.
“I was a young man with unformed ideas,” he’d said. “I threw out queries, suggestions, wondering all the time over everything, and to my astonishment, the ideas took like wildfire. People made a religion of them.” He begged her to preach to a gathering at his summer house the following day. “If you take the meeting at three o’clock this window will be open, and you will know that I am joining in with the singing.”
It’s not impossible that Cotton did meet Darwin, but his family denied that he had ever recanted the theory. “Dear Sir,” he had written in an 1880 letter, “I am sorry to inform you that I do not believe in the Bible as a divine revelation, & therefore not in Jesus Christ as the son of God.”
Math Notes
19 can be expressed as the sum of four perfect cubes:
19 = 333 + (-29)3 + (-25)3 + 163
So can 6941:
6941 = (-3476)3 + 30903 + 23173 + 3843
Can every integer be expressed in this way? It’s conjectured that the answer is yes, but no one has yet managed to prove it.
Verse
This poem was composed by English mathematician George Biddell Airy, the seventh Astronomer Royal. It’s written from the point of view of the irregular polygon in the middle of the figure. If two congruent right triangles are affixed to the bottom edges of this polygon, the resulting shape has the outline of two adjoining squares of side lengths a and b, the sides of the right triangles. If the same two triangles are instead affixed above the polygon, as shown, the result is a (tilted) square whose side is the length of the triangles’ hypotenuse. This proves the Pythagorean theorem.
“Various proofs come near to this, but none which I can find comes up to it in simplicity,” wrote Augustus De Morgan to William Rowan Hamilton in 1855. “Placing the triangle in four different positions shows the proposition anatomically, without scalpel or nasty smell.”
The Sylvester–Gallai Theorem
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.
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.)
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.)
Quinary
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:
