In the early 1900s, blue tits and robins had easy access to cream from the tops of open milk bottles left on humans’ doorsteps. After World War I, the humans began to seal the bottle tops with aluminum foil. But remarkably, by the 1950s the entire blue tit population of the United Kingdom had learned pierce the foil to reach the cream, while the robins hadn’t.
The difference lay in cultural transmission: A blue tit can learn a new behavior by observing another bird performing it. Robins generally can’t do this — while an individual robin might learn to pierce the foil, it has no way to pass on this discovery to other robins. Young blue tits are reared in flocks in which they can observe one another, which is an advantage; robins are territorial and have fewer such opportunities.
Suppose s is the infinite series 1 – 2 + 3 – 4 + …. The diagram above presents 4 copies of the series. Each white disk represents +1, and each red disk represents -1. Each pair of red and white disks annihilates to zero, and the connecting lines show that all of the disks beyond the first +1 (green) can be paired off in this way. The result is that 4s = 1 and, perversely, that 1 – 2 + 3 – 4 + … = 1/4.
(It’s not really that simple — this series doesn’t tend toward any finite limit, but any summation method that’s linear and stable does produce the sum 1/4.)
In 1902, chemist Harvey Wiley launched a unique experiment to test the safety of food additives. He recruited a group of young men and fed them meals laced with chemicals to see what the effects might be. In this week’s episode of the Futility Closet podcast we’ll describe Wiley’s “poison squad” and his lifelong crusade for food safety.
We’ll also follow some garden paths and puzzle over some unwelcome weight-loss news.
Inspired by Edwin Abbott’s Flatland, in which a three-dimensional sphere tries to explain its world to a two-dimensional square, Worcester Polytechnic Institute physicist P.K. Aravind has devised a ballet that describes a tesseract, or four-dimensional hypercube, to a three-dimensional audience.
“The spirit of my demonstration is very similar to Abbott’s, only it is pitched at Spacelanders who are encouraged to make the leap from three dimensions to four, just as Abbott’s demonstration was pitched at a Flatlander who was encouraged to make the leap from two dimensions to three.”
This is a thrackle, a set of lines connecting points such that each two lines intersect exactly once (if two lines have a common endpoint, that counts as an intersection).
Princeton mathematician John Conway asks: Can a thrackle have more lines than points? He will pay $1,000 for a solution; so far no one’s managed to find one.
Jazz guitarist Pat Martino had a burgeoning record career by age 20, but in 1976 he began to suffer headaches, followed by mania, depression, and seizures. He attempted suicide several times, but hospitalization and electroshock therapy brought no relief. In 1980 a CT scan discovered an arteriovenous malformation that had begun to hemorrhage, and a surgeon removed 70 percent of Martino’s left temporal lobe.
After the surgery he didn’t know his name, recognize his parents, or know he was a musician. When his father played his old records for him, “I would lie in my bed upstairs and hear them seep through the walls and the floor, a reminder of something that I had no idea that I was supposed to be anymore, or that I ever was.” But when a visiting friend played a major seventh chord, Martino found that he wanted a minor ninth and took up the instrument again.
“As I continued to work out things on the instrument, flashes of memory and muscle memory would gradually come flooding back to me — shapes on the fingerboard, different stairways to different rooms in the house,” he wrote.
Aided by his father, friends, photographs, and mainly by his own recordings, he learned the instrument afresh, “to escape the situation, and to please my father.” Neurosurgeon Marcelo Galarza writes, “The process of memory retrieval took him about two years. Although he never lost his manual dexterity, the necessary skill to play guitar again to his previous musical level took years to bring back.”
In 1987 he recorded his comeback album, The Return, and he’s made more than 20 albums since then. Galarza writes, “To our knowledge, this case study represents the first clinical observation of a patient who exhibited complete recovery from a profound amnesia and regained his previous virtuoso status.”
In 1959, when the world was casting about for peaceful applications of fission energy, activist Muriel Howorth established the Atomic Gardening Society, a global group of amateur gardeners who cultivated irradiated seeds, hoping for useful mutations. Howorth published a book, Atomic Gardening for the Layman, and crowdsourced her effort, distributing seeds to her members and collating their results. She herself made news with “the first atomic peanut,” a 2-foot-tall peanut plant that had sprouted from an irradiated nut.
She teamed up with Tennessee dentist C.J. Speas, who had a license for a cobalt-60 source and had built a cinderblock bunker in his backyard. Via Howorth he distributed millions of seeds to thousands of society members, but the odds remained against them: It would likely require many times this number to hit on a mutation that was potentially useful.
The Atomic Gardening Society disbanded within a few years, but it gave way to more ambitious “gamma gardens” of 5 acres and more in which plants are arranged in rings around a central radiation source. This technique continues today in America and Japan.
Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously.
Kenya and Uganda both lie on the equator, so the sun rises around 6 a.m. and sets around 6 p.m. throughout the year. Given such a reliable natural timekeeper, it’s customary to reckon time by counting hours of light or hours of darkness: 7 a.m. is called 1 o’clock (saa moja, or one hour of light), and 11 a.m. is called 5 o’clock (saa tano) (moja means 1 and tano 5 in Swahili). Similarly, 7 p.m. is called 1 o’clock (one hour of darkness), and 11 p.m. is 5 o’clock.
Confusingly for newcomers, clocks themselves are set to Western time, but they’re read aloud in “Swahili time.” Increasingly, though, Africans are simply conforming to Western conventions.
Take any two rational numbers whose product is 2, and add 2 to each. The results are the legs of a right triangle with rational sides.
For example, 13/17 × 34/13 = 2. If we add 2 to each of these we get 47/17 and 60/13 or, clearing fractions, 611 and 1020. The hypotenuse is 1189.
Because (z + 2)2 + [(2/z) + 2]2 = [z + 2 + (2/z)]2, if z is rational, so are all three sides.
(R.S. Williamson, “A Formula for Rational Right-Angle Triangles,” Mathematical Gazette 37:322 [December 1953], 289-290, via Claudi Alsina and Roger B. Nelsen, Icons of Mathematics, 2011.)
.jpg){kind=link}