Two for One

https://commons.wikimedia.org/wiki/File:Doom_Mons,_Sotra_Patera,_Mohini_Fluctus_(cropped).png

Mountains on Saturn’s moon Titan are named after mountains in Middle-earth, the fictional setting of J.R.R. Tolkien’s fantasy novels.

The highest peak on Titan is Mount Doom (“Doom Mons”), which rises more than a mile above the surrounding plain. Tolkien’s Mount Doom made its first appearance in The Lord of the Rings in 1954.

By coincidence, science fiction writer Stanley G. Weinbaum had already placed a fictional Mount Doom on Titan in his 1935 story Flight on Titan.

So, in honoring Tolkien, the International Astronomical Union also fulfilled Weinbaum’s vision.

Equilibrium

https://commons.wikimedia.org/wiki/File:StevinEquilibrium.svg

Drape a chain of evenly spaced weights over a pair of (frictionless) inclined planes like this. What will happen? There’s more mass on the left side, but the slope on the right side is steeper. Simon Stevin (1548-1620) realized that in fact the chain won’t move at all — if it did, we could link the ends as shown and produce a perpetual motion machine.

This is remembered as the “Epitaph of Stevinus.” Richard Feynman wrote, “If you get an epitaph like that on your gravestone, you are doing fine.”

Hope and Change

u.s. coins

The denominations of U.S. coins make intuitive sense, but they can be unwieldy: It can take up to eight coins to assemble an amount up through 99¢. Indeed, producing 99¢ takes (1 × 50¢) + (1 × 25¢) + (2 × 10¢) + (4 × 1¢). What five denominations would minimize the number of coins ever needed to make change?

In The Math Chat Book, Frank Morgan reports that with coins of 1¢, 3¢, 11¢, 27¢, and 34¢, you never need more than 5 coins to make change. For example, now 99¢ = (2 × 34¢) + (1 × 27¢) + (1 × 3¢) + (1 × 1¢). Of the 1,129 possible solutions, this one requires the fewest coins on average (3.343).

Unfortunately, this system is a bit tricky too — to assemble some totals, it’s more efficient to use a few middle-size coins rather than starting with the largest value possible. For example, if you assemble 54¢ by starting with a 34¢ coin, it takes four additional coins to gather the remaining 20¢: (1 × 11¢) + (3 × 3¢). It would have been simpler to choose 2 × 27¢, but that’s not immediately evident.

Turning Point

https://commons.wikimedia.org/wiki/File:Pythagoras_theorem_leonardo_da_vinci.png
Image: Wikimedia Commons

This pretty proof of the Pythagorean theorem is attributed to Leonardo da Vinci. Draw a right triangle and construct a square on each side, and make a copy of the original triangle and add it to the bottom of the hypotenuse square as shown. Now the shaded hexagon in the first figure can be rotated 90 degrees clockwise around the indicated point to occupy the position shown in the second figure. The orange and green quadrilaterals in the second figure are seen to be congruent to those in the first figure: The three shortest sides of the orange quadrilateral in the second figure correspond to their counterparts in the first, and the angles between them are assembled from the same constituents. The same is true of the green quadrilaterals. In each figure the shaded hexagon contains two instances of the original right triangle; remove these and we can see that the two squares in the first figure equal the large square in the second figure, proving Pythagoras.

10/10/2021 UPDATE: A number of readers point out that only the orange quadrilateral here can properly be said to turn; in the second diagram the green quadrilateral has been reflected as well. (Thanks, Mark and Bill.)

Order

In a December 1985 letter to the Mathematical Gazette, Middlesex Polytechnic mathematician Ivor Grattan-Guinness writes that Astronomer Royal George Biddell Airy “would sometimes go around the Observatory, and on finding an empty box, insert a piece of paper saying ‘Empty box’ and thereby falsify its description! This last achievement deserves, in my proposal, the name of ‘Airy’s paradox’.”

A Geometric Illusion

geometric illusion

Which of the two shaded areas is larger, the central disc or the outer ring?

Surprisingly, they’re equal. Each of the concentric circles has a radius 1 unit larger than the last. So the area of the central disc is π × 32 square units, and the area of the outer ring is π × 52 – π × 42 = π × 32 square units. So the two areas are the same.

A Cool Customer

A brewery stored its beer in a cellar some distance from the bottling plant. The cellar was cooled by pipes that circulated a saline solution from a central cooling unit. The main pipe that connected this cooling unit and the cellar happened to pass near the cellar of a retailer.

The brewery’s owner eventually discovered that the retailer was using the saline solution to cool his own cellar. He sued the retailer for theft, but the judge ruled, “In accordance with Article 242 of the Criminal Code, theft is the unlawful appropriation of commodities belonging to another party. In the present case no theft has been committed, since the saline solution was not misappropriated; rather, it was returned in its entirety to the brewery’s main pipe.”

The brewery owner appealed the case, arguing, “The issue is not the theft of saline solution but the theft of energy. If the saline solution is used to cool the defendant’s cellar in addition to my own, I have to pay more for electricity to operate the central cooling unit.”

The court of appeals ruled: “The saline solution acquires heat from the retailer’s cellar; therefore, energy belonging to the brewery is not being stolen. On the contrary, the brewery is receiving gratuitous energy from the retailer.”

This story appeared in a German scientific monograph, “Questions of Thermodynamical Analysis,” by P. Grassman. In propounding it in May 1990, Quantum added, “We all agree the judge was wrong, but not everyone can correctly explain his error. Can you?”

The Grim Reaper Paradox

https://commons.wikimedia.org/wiki/File:Death-and-the-woodcutter-jean-francois-millet3.jpg

Suppose there are an infinite number of Grim Reapers. Each has an appointed time to kill Fred if it finds him alive.

The last Grim Reaper (call it #1) is appointed to do this exactly one minute after noon. The next-to-last (#2) is appointed to do it one half minute after noon. And so on: If it finds him alive, Reaper n will kill Fred exactly 1/2(n-1) minutes after noon.

Thus there is no first Reaper. For any given Reaper, there are infinitely many others who precede it by moments.

Whatever happens, we know that Fred can’t survive this ordeal — to go on with his life he must still be alive at 12:01, and we know for certain that if he lives that long then Reaper #1 will kill him. But in order to survive to 12:01 he must still be alive at 30 seconds after 12 — and at that time Reaper #2 will kill him. And so on. It appears that no Reaper will ever get the chance to kill Fred, because each is preceded by another who will rob him of the opportunity.

So it’s impossible that Fred survives, but it’s also impossible that any Reaper kills him. Must we say that he dies for certain but of no cause?

(From José Benardete’s Infinity: An Essay in Metaphysics, 1964.)

Youth and Genius

Mathematician Norbert Wiener entered university at age 11 and earned a doctorate at 17, but he was 7 years old before he learned that Santa Claus does not exist. From his 1953 memoir Ex-Prodigy:

“At that time I was already reading books of more than slight difficulty, and it seemed to my parents that a child who was doing this should have no difficulty in discarding what to them was obviously a sentimental fiction. What they did not realize was the fragmentariness of the child’s world.”

In his 1909 autobiography Memories of My Life, Francis Galton remembers a boarding school to which he was sent at age 8:

“In that room was a wardrobe full of schoolbooks ready for issue. It is some measure of the then naïveté of my mind that I wondered for long how the books could have been kept so fresh and clean for nearly two thousand years, thinking that the copies of Caesar’s Commentaries were contemporary with Caesar himself.”

In Fragments of Genius, his 1989 survey of the feats of idiots savants, Michael Howe notes that a study of 8-year-olds who were exceptional chess players showed that they were perfectly normal in other spheres. “And the transcripts of interviews in which highly gifted young adults talk about their childhoods, supplemented by interviews with their parents, are full of testimonies to the extreme ordinariness of the individuals, outside their particular area of special talent.”

Credit

“The first author would like to acknowledge and thank Jesus Christ, through whom all things were made, for the encouragement, inspiration, and occasional hint that were necessary to complete this article. The second author, however, specifically disclaims this acknowledgement.”

— Michael I. Hartley and Dimitri Leemans, “Quotients of a Universal Locally Projective Polytope of Type {5, 3, 5},” Mathematische Zeitschrift 247:4 (2004), 663-674