During World War I, Ernest Rutherford worked tirelessly on a secret project to detect submarines by sonar. But on one occasion he did decline to attend a committee meeting.
“I have been engaged in experiments which suggest that the atom can be artificially disintegrated,” he wrote. “If it is true it is of far greater importance than a war.”
In 1959 chemist William J. Buehler of the Naval Ordnance Laboratory was trying to devise a missile nose cone that could withstand extraordinary heat and fatigue. He found a promising alloy of nickel and titanium and passed around a sample at a 1961 laboratory management meeting. The sample had been folded like an accordion, but in examining it Buehler’s colleagues flexed and twisted it out of shape. When of them idly held it over his pipe lighter, they got a surprise: The sample sorted itself back into its accordion shape.
Buehler’s alloy is now known as nitinol (for “nickel titanium Naval Ordnance Laboratory”), and this property is known as “shape memory.” In Nature’s Building Blocks, John Emsley notes, “Spectacle frames made from nitinol can be bent and twisted into remarkable shapes and, when released, will jump back to their original shape.”
Here’s an odd little animal: Get two rigid disks, cut a notch in each one, fit them together as shown, and try to send them rolling across a table. If the notches are too deep, marrying the discs too closely together, then the object will pretty quickly slow to a stop with each disc standing at a 45° angle to the table. If the notches are too shallow, it will stop with one disc standing up at right angles to the table. But if the notches are about the right length, ideally 29.2893 percent of the radius, then the contraption will roll along quite happily for a surprisingly long distance.
The reason is that in that configuration the object’s center of mass remains level as it rolls along. (It does move from side to side, which is why it’s called the wobbler.)
Apparently this was originally discovered by A.T. Stewart, who dubbed his creation the “two-circle roller” in a 1966 note in the American Journal of Physics. I found it described in Matt Parker’s 2014 book Things to Make and Do in the Fourth Dimension, which includes a simple proof of the principle involved. There’s a more rigorous discussion here.
How many animals of each kind did Moses take on the ark? If you’re like most people, you tend to answer two, even though you know it was Noah, not Moses, who took animals on the ark. People tend to have difficulty noticing when a term in a sentence or question is replaced with a semantically similar but incorrect term.
This isn’t really surprising on its face. What’s surprising is how robust the effect is. About 50 percent of people make the mistake even when asked to read the question aloud before answering it. The effect persists even when people are warned that a distortion might be present, and most people express confidence in their answer even when given unlimited time to think about it. Further examples:
One possible explanation is “partial matching” — the distorted question so closely resembles one that we recognize that we take the risk of jumping to the answer. “Everyday cognitive processing must be based on simple heuristics such as matching sets of features rather than exact matches, as very few tasks require exact matches,” suggest researchers Heekyeong Park and Lynne M. Reder. “Partial matching is immutable because it is the most efficient way for memory to operate, given the nature of the environment in which we live.”
(Heekyeong Park and Lynne M. Reder, “Moses Illusion,” in Rüdiger F. Pohl, ed., Cognitive Illusions, 2004.)
Engaged to give a talk at a university, logician Raymond Smullyan arrived half an hour early and wrote the following sentence on the blackboard, “to give the audience something to mull over”:
You have no reason to believe this sentence.
This, he reasoned, was a paradox. If you have no reason to believe the sentence, then what it states is really the case, which is certainly a good reason to believe it. But if you have a good reason to believe it, then it must be true … which means that you have no reason to believe it.
Half an hour later he came down the stairs to a packed audience. Spotting a bright-looking boy in the front row, he pointed to the sentence and asked him, “Do you believe that sentence?”
“Yes,” said the boy.
“What is your reason?”
“I don’t have any.”
Smullyan asked, “Then why do you believe it?”
The boy said, “Intuition.”
(Raymond Smullyan, “Self-Reference in All Its Glory!” conference “Self-Reference,” Copenhagen, Oct. 31-Nov. 2, 2002.)
The samurai crab, Heikea japonica, earns its nickname well: Its shell bears a startling resemblance to the face of an angry warrior. Some Japanese believe that these crabs are reincarnated samurai who, defeated at the Battle of Dan-no-ura, threw themselves into the sea, as described in the epic Tale of the Heike.
Biologist Julian Huxley put forward the idea that the “faces” were an example of artificial selection. He suggested that fishermen who caught crabs with particularly face-shaped carapaces, believing them to be reincarnated spirits, threw them back into the sea, permitting them to reproduce while their brothers were eaten.
But humans don’t eat these crabs, and in any case the “warrior” crabs exist even far from sites of human fishing. Really the crabs are an example of another, equally compelling phenomenon — pareidolia, our tendency to see significant patterns where none exist.
This is Charles Darwin’s first diagram of an evolutionary tree, from his First Notebook on Transmutation of Species. He drew it around July 1837, barely a month after he’d opened his first full transmutation notebook.
“Case must be that one generation should have as many living as now,” he wrote. “To do this and to have as many species in same genus (as is) requires extinction. Thus between A + B the immense gap of relation. C + B the finest gradation. B + D rather greater distinction. Thus genera would be formed. Bearing relation to ancient types with several extinct forms.”
At the top he’s written “I think.”
The Shepard tone is an auditory illusion: A succession of overlapping scales are played, each ascending, and each scale fades out as its successor fades in an octave lower. The resulting impression is of a climbing pitch that never “arrives” anywhere, a rising note that never gets higher.
Among many other applications, this sound was used for the Batpod in Christopher Nolan’s films The Dark Knight and The Dark Knight Rises — the vehicle seems constantly to accelerate without ever changing gear. “When played on a keyboard,” wrote sound designer Richard King, “it gives the illusion of greater and greater speed; the pod appears unstoppable.”
(Thanks, Nick.)
01/31/2022 UPDATE: Similarly, the Risset Rhythm seems to speed up:
(Thanks, Chris.)
Here’s something remarkable: This formula, discovered in 1995 by David Bailey, Peter Borwein, and Simon Plouffe of the University of Quebec at Montreal, permits the calculation of isolated digits of π — it’s possible to calculate, say, the trillionth digit of π without working out all the preceding digits.
The catch is that it works only in base 2 (binary) and base 16 (hexadecimal), but not in base 10. So it’s possible to know, say, that the five trillionth binary digit of π is 0, but there’s no way to convert the result into its decimal equivalent without working out all the intervening binary digits.
“The new formula allows the calculation of the nth base 2 or base 16 digit of π in a time that is essentially linear in n, with memory requirements that grow logarithmically (very slowly) in n,” writes David Darling in The Universal Book of Mathematics. “One possible use of the Bailey-Borwein-Plouffe formula is to help shed light on whether the distribution of π’s digits are truly random, as most mathematicians suppose.”
08/14/2022 UPDATE: A new formula permits the extraction of decimal digits. (Thanks, Edward.)
A puzzle from reader Steven Moore:
Find A, B, and C as distinct integers. There is only one solution.
