Two Dire Punishments

Under Roman law, subjects found guilty of patricide were subjected to poena cullei, the “penalty of the sack” — they were sewn into a leather sack with a snake, a cock, a monkey, and a dog and thrown into water.

In his Life of Artaxerxes, Plutarch describes an ancient Persian method of execution known as scaphism in which vermin devour a victim trapped between mated boats:

Taking two boats framed exactly to fit and answer each other, they lie down in one of them the malefactor that suffers, upon his back; then, covering it with the other, and so setting them together that the head, hands, and feet of him are left outside, and the rest of his body lies shut up within, then forcing him to ingest a mixture of milk and honey before pouring all over his face and body. They then keep his face continually turned towards the sun; and it becomes completely covered up and hidden by the multitude of flies that settle on it. And as within the boats he does what those that eat and drink must needs do, creeping things and vermin spring out of the corruption and rottenness of the excrement, and these entering into the bowels of him, his body is consumed.

Happily Plutarch seems to have based his account on a report by the Greek historian Ctesias, whose reliability has been questioned, so perhaps this never happened.

All Right Then

In the 1970’s, a student of Maoist inclination asked [Columbia philosopher Sidney Morgenbesser] if he disagreed with Chairman Mao’s saying that a proposition can be true or false at the same time. Dr. Morgenbesser replied, ‘I do and I don’t.’

From Morgenbesser’s 2004 New York Times obituary.

October 3, 2024 | Science & Math

Unquote

“I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops.” — Stephen Jay Gould, The Panda’s Thumb, 1980

October 2, 2024October 1, 2024 | Quotations

Four Glasses

Martin Gardner published this puzzle in his “Mathematical Games” column in Scientific American in February 1979. You’re blindfolded and sitting before a lazy susan. On each corner is a glass. Some are right side up and some upside down. On each turn you can inspect any two glasses and, if you choose, reverse the orientation of either or both of them. After each turn the lazy susan will be rotated through a random angle. When all four glasses have the same orientation, a bell will sound. How can you reach this goal in a finite number of turns?

	Select Click for Answer
This algorithm will do it in five turns: Choose a pair of glasses that are diagonally opposite and set both right side up. Choose two adjacent glasses. At least one must now be right side up. If the other is upside down, turn it right side up as well. If the bell rings, you’re done; if not, then only one glass is still upside down. Choose a diagonally opposite pair. If one is upside down, turn it up and you’re done. If both are right side up, turn one upside down. Now two glasses are upside down, and these two must be adjacent. Choose two adjacent glasses and reverse the orientation of both. If the bell doesn’t sound then there are now two glasses upside down, and they’re diagonally opposite one another. Choose a diagonally opposite pair and reverse both. The bell will sound.

Click for Answer

October 2, 2024October 1, 2024 | Puzzles

Cameo

Grip, the talking raven in Dickens’ Barnaby Rudge, was based on a real bird, a pet who lived in the family’s Marylebone home, where she buried items in the garden, terrorized the dog, bit the children, and tore at the family carriage. She died in 1841 after ingesting some lead-based paint, and Dickens wrote her into the novel, which appeared later that year.

Interestingly, when Edgar Allan Poe reviewed the story for Graham’s Magazine, he remarked that “The raven … might have been made more than we see it … Its croaking might have been prophetically heard in the course of the drama.” In 1842, when the two authors met in Philadelphia, Poe was said to be “delighted” that Grip had been based on a real bird.

Poe’s famous poem appeared three years later, and scholars generally agree that Grip had inspired the “ebony bird” — in Dickens’ novel the raven had repeated the phrases “Never say die” and “Nobody” and is described as “tapping at the door” and “knocking softly at the shutter.”

Today Grip’s remains are on display in Philadelphia’s Parkway Central Library, where they may inspire yet more writings.

October 1, 2024September 30, 2024 | Literature

Noted

https://commons.wikimedia.org/wiki/File:Toronto_recursive_history_plaque.jpg — Image: Wikimedia Commons

October 1, 2024September 30, 2024 | Humor

Band Practice

Drill a hole straight through the center of a sphere, leaving a band in the shape of a napkin ring. Suppose the height of this band is h. Now take a sphere of a different size and drill a hole through that, contriving the hole’s width so that its depth is again h. Remarkably, the two napkin rings will have the same volume.

As the sphere expands, the band must grow both wider and thinner, and it turns out that these two effects exactly cancel one another. It’s called the napkin ring problem.

September 30, 2024September 29, 2024 | Science & Math

Passing Through

Just a striking image: German astronomer Johann Friedrich Julius Schmidt prepared a plaster moon for the Field Columbian Museum in 1898. From the Field Museum Library.

September 30, 2024September 29, 2024 | Science & Math

Flat Devotion

William Linkhaw sang so badly that a grand jury indicted him for disrupting his church’s services. At trial in August 1872, a witness imitated Linkhaw’s singing style and provoked “a burst of prolonged and irresistible laughter, convulsing alike the spectators, the Bar, the jury and the Court.”

It was in evidence that the disturbance occasioned by defendant’s singing was decided and serious; the effect of it was to make one part of the congregation laugh and the other mad; that the irreligious and frivolous enjoyed it as fun, while the serious and devout were indignant.

Linkhaw protested that he felt a duty to worship God. The jury fined him a penny, but the North Carolina Supreme Court set aside the verdict, observing that Linkhaw had had no malicious intent. Justice Thomas Settle wrote, “It would seem that the defendant is a proper subject for the discipline of his church, but not for the discipline of the Courts.”

In 1906 a wit wrote, “although the proof did show / That Linkhaw’s voice was awful / The judges found no valid ground / For holding it unlawful.”

September 29, 2024September 28, 2024 | Oddities

A Pi Diet

Students beginning with the compass learn to draw this rosette, sometimes called the Flower of Life. If the arcs and the circle have the same radius, a, then the area of one petal is

$\displaystyle X = a^{2}\left ( \frac{\pi }{3} - sin \frac{\pi }{3}\right ) = \left ( \frac{\pi }{3} - \frac{\sqrt{3}}{2}\right ) a^{2}$

and the unshaded area of the circle is

$\displaystyle Y = \pi a^{2} - 3X = \frac{3\sqrt{3}}{2}a^{2}.$

Remarkably, though the area we sought is bounded entirely by the arcs of circles, the final expression is independent of π.

Related: When two cylinders of radius r meet at right angles, the volume of their intersection is 16r³/3 — again, no sign of π.

(J.V. Narlikar, “A Pi-Less Area,” Mathematical Gazette 65:431 [March 1981], 32-33.)

10/01/2024 UPDATE: Some deft rearranging shows that the unshaded area of the circle is just the area of an inscribed regular hexagon:

2024-09-28-a-pi-diet-2

So the absence of π isn’t that surprising. Thanks to readers Catalin Voinescu (who sent this diagram) and Gareth McCaughan for pointing this out.

September 28, 2024October 1, 2024 | Science & Math