In 1938, Italian physicist Ettore Majorana vanished after taking a sudden sea journey. At first it was feared that he’d ended his life, but the perplexing circumstances left the truth uncertain. In this week’s episode of the Futility Closet podcast we’ll review the facts of Majorana’s disappearance, its meaning for physics, and a surprising modern postscript.
We’ll also dither over pronunciation and puzzle over why it will take three days to catch a murderer.
Statistics textbooks sometimes ask: Suppose you’re driving on the highway and adjust your speed so that the number of cars you pass is equal to the number that pass you. Is your speed the median or the mean speed of the cars on the highway?
The expected answer is that it’s the median speed, since the number of cars traveling more slowly than you is equal to the number traveling faster. But California State University mathematician Larry Clevenson and his colleagues wrote in 2001, “This certainly is true of the cars that you see, but that isn’t what the problem asks, and it isn’t the correct answer.”
Surprisingly, they found that the correct answer is the mean. “If you adjust your speed so that as many cars pass you as you pass, then your speed is the mean speed of all the other cars on the highway.” Details at the link below.
(Larry Clevenson et al., “The Average Speed on the Highway,” College Mathematics Journal 32:3 [2001], 169-171.)
Temple University anthropologist Wayne Zachary was studying a local karate club in the early 1970s when a disagreement arose between the club’s instructor and an administrator, dividing the club’s 34 members into two factions. Thanks to his study of communication flow among the members, Zachary was able to predict correctly, with one exception, which side each member would take in the dispute.
The episode has become a popular example in discussions of community structure in networks, so much so that scientists now award a trophy to the first person to use it at a conference. The original example is known as Zachary’s Karate Club; the trophy winners are the Zachary’s Karate Club Club.
(Wayne W. Zachary, “An Information Flow Model for Conflict and Fission in Small Groups,” Journal of Anthropological Research 33:4 [1977], 452-473. Thanks to Snehal Shekatkar for the image.)
09/01/2024 Reader Peter Dawyndt points out that the reason for the single exception in Zachary’s prediction is notable. The person whom Zachary assigned to the wrong faction corresponds to node 9 in this graph of the network:
“This person joined the newly founded karate club with supporters of the teacher (node 1) after the split, despite being a weak supporter of the president (node 34). This choice stemmed mainly from opportunism: he was only three weeks away from a test for black belt (master status) when the split in the club occurred. Had he joined the president’s club, he would have had to give up his rank and begin again in a new style of karate with a white (beginner’s) belt, since the president had decided to change the style of karate practiced in his club. Having four years of study invested in the style of the original club’s instructor, the individual could not bring himself to repudiate his rank and start again.”
(Thanks, Peter.)
This cottage, at 112 Mercer Street in Princeton, New Jersey, has been home to three Nobel winners: Albert Einstein lived there from 1935 to 1955; physicist Frank Wilczek between 1989 and 2001; and economist Eric Maskin until 2012.
It resides on the National Register of Historic Places and has been designated a U.S. National Historic Landmark, but it bears no outward marker of its significance.
In The Art of Computer Programming, Donald Knuth notes an interesting pattern in the common units of liquid measure in 13th-century England:
2 gills = 1 chopin
2 chopins = 1 pint
2 pints = 1 quart
2 quarts = 1 pottle
2 pottles = 1 gallon
2 gallons = 1 peck
2 pecks = 1 demibushel
2 demibushels = 1 bushel or firkin
2 firkins = 1 kilderkin
2 kilderkins = 1 barrel
2 barrels = 1 hogshead
2 hogsheads = 1 pipe
2 pipes = 1 tun
“Quantities of liquid expressed in gallons, pottles, quarts, pints, etc. were essentially written in binary notation,” Knuth writes. “Perhaps the true inventors of binary arithmetic were British wine merchants!”
(Thanks, Colin.)
What most clinicians do when they receive a laboratory report is, of course, to look up the normal range for the tests in question. … Traditionally, a normal range is calculated in such a way that it includes 95% of the results found in a group of normal or healthy persons, and, consequently, there is a 5% risk that a healthy person will present with an abnormal laboratory result. Then, imagine that you do ten tests on a normal person. In that case the risk that at least one of these tests is abnormal is (1 – 0.9510) which amounts to 0.40 or 40%. If you do twenty-five tests (and that is not uusual in clinical practice), this chance is 72%! As Edmond A. Murphy puts it so aptly, ‘Therefore, a normal person is anyone who has not been sufficiently investigated.’
— Henrik R. Wulff, Stig Andur Pedersen, and Raben Rosenberg, Philosophy of Medicine: An Introduction, 1990, citing Murphy’s The Logic of Medicine, 1976
Spiders are largely blind but experience the world through vibrations in their webs, which they detect with their legs. Now MIT materials scientist Markus Buehler has created a musical impression of that experience. His team made three-dimensional maps of the webs of tropical tent-web spiders, identified the vibrating frequency of each thread, and converted these frequencies into ranges that humans can hear. Threads that are closer to the spider sound louder in the recording.
“The spider web can be viewed as an extension of the body of the spider, in that it lives within it, but also uses it as a sensor,” Buehler told New Scientist. “When you go into the virtual reality world and you dive inside the web, being able to hear what’s going on allows you to understand what you see.”
(Thanks, Colin.)
Leonardo noted that studies of shadow appeal to the same geometric laws as those of perspective — “in matters perspectival, a light source is no different from the eye [… since] the visual ray is similar to the shadowy ray in terms of speed and converging lines.” Just as perspective considers a scene as regarded from a particular point of view, a shadow is a record of what “the lamp cannot see.”
Philosopher Roberto Casati writes, “This is a strong analogy. There is shadow where the light cannot see. Indeed, shadow is exactly what the light source cannot see. A lamp sees only the things that it illuminates; objects in shadow are behind lighted objects that screen the shadow from the lamp’s perspective. For this same reason, when we examine a shadow, we discover the profile of things from the point of view of the lamp. … Shadow allows us to see a point of view other than our own without even getting up from our seats.”
(From Casati’s Shadows, 2000, via Marko Uršič, Shadows of Being: Four Philosophical Essays, 2019.)
A bedeviling little puzzle from Dickinson College mathematician David Richeson.
A topologist, it is said, is someone who can’t tell his donut from his coffee mug.
Before the 19th century, containers did not come in standard sizes, and students in the 1400s were taught to “gauge” their capacity as part of their standard mathematical education:
There is a barrel, each of its ends being 2 bracci in diameter; the diameter at its bung is 2 1/4 bracci and halfway between bung and end it is 2 2/9 bracci. The barrel is 2 bracci long. What is its cubic measure?
This is like a pair of truncated cones. Square the diameter at the ends: 2 × 2 = 4. Then square the median diameter 2 2/9 × 2 2/9 = 4 76/81. Add them together: 8 76/81. Multiply 2 × 2 2/9 = 4 4/9. Add this to 8 76/81 = 13 31/81. Divide by 3 = 4 112/243 … Now square 2 1/4 = 2 1/4 × 2 1/4 = 5 1/16. Add it to the square of the median diameter: 5 5/16 + 4 76/81 = 10 1/129. Multiply 2 2/9 × 2 1/4 = 5. Add this to the previous sum: 15 1/129. Divide by 3: 5 1/3888. Add it to the first result: 4 112/243 + 5 1/3888 = 9 1792/3888. Multiply this by 11 and then divide by 14 [i.e. multiply by π/4]: the final result is 7 23600/54432. This is the cubic measure of the barrel.
Interestingly, this practice informed the art of the time — this exercise is from a mathematical handbook for merchants by Piero della Francesca, the Renaissance painter. Because many artists had attended the same lay schools as business people, they could invoke the same mathematical training in their work, and visual references that recalled these skills became a way to appeal to an educated audience. “The literate public had these same geometrical skills to look at pictures with,” writes art historian Michael Baxandall. “It was a medium in which they were equipped to make discriminations, and the painters knew this.”
(Michael Baxandall, Painting and Experience in Fifteenth Century Italy, 1988.)
04/10/2021 UPDATE: A reader suggests that there’s a typo in the original reference here. If 9 1792/3888 is changed to 9 1793/3888, the final result is 7 23611/54432, which is exactly the result obtained by integration using the approximation π = 22/7. (Thanks, Mariano.)
