The Hypercubical Dance

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.”

He describes the project here.

(P.K. Aravind, “The Hypercubical Dance — A Solution to Abbott’s Problem in Flatland?,” Mathematical Gazette 91:521 [July 2007]: 193-197.)

Image: Flickr

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.”

(Marcelo Galarza et al., “Jazz, Guitar, and Neurosurgery: The Pat Martino Case Report,” World Neurosurgery 81:3 [2014], 651-e1.)

Atomic Gardening
Image: Wikimedia Commons

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.

Chapter One

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.

— David Goodstein, States of Matter, 1985

Swahili Time
Image: Wikimedia Commons

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.

The Right Stuff

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.)

The Map of Discernment

In To Predict Is Not to Explain, mathematician René Thom describes a lunch at which psychoanalyst Jacques Lacan responded strongly to his statement that “Truth is not limited by falsity, but by insignificance.” Thom described the idea later in a drawing:

At the base, one finds an ocean, the Sea of the Insignificance. On the continent, Truth is on one side, Falsehood on the other. They are separated by a river, the River of Discernment. It is indeed the faculty of discernment that separates truth from falsehood. It’s Aristotle’s notion: the capacity for contradiction. It’s what separates us from animals: When information is received by them, it’s instantly accepted and it triggers obedience to its message. Human beings, however, have the capacity to withdraw and to question its veracity.

Following the banks of this river, which flows into the Sea of Insignificance, one travels along a coastline that is slightly concave: Situated at one end is the Slough of Ambiguity; at the other end is the Swamp of La Palice. At the head of the river delta, one sees the Stronghold of Tautology: That’s the stronghold of the logicians. One climbs a rampart towards a small temple, a kind of Parthenon: that’s Mathematics.

To the right, one finds the Exact Sciences: Up in the mountains that surround the bay is Astronomy, with an observatory topping its temple; at the far right stand the giant machines of Physics, the accelerator rings at CERN; the animals in their cages indicate the laboratories of Biology. Out of all this, there emerges a creek that feeds into the Torrent of Experimental Science, which flows into the Sea of Insignificance.

To the left is a wide path climbing towards the north west, up to the City of Human Arts and Sciences. Continuing along it one comes to the foothills of Myth. We’ve entered the kingdom of anthropology. Up at the top is the High Plateau of the Absurd. The spine signifies the loss of the ability to discern contraries, something like an excess of universal understanding which makes life impossible.

He explains the central idea in more detail starting on page 173 of the PDF linked above. “It’s something I’ve done to amuse myself, but it reflects something real, I think: The Logos, the possibility of representation by language, only comes into play for humanity in a rather limited number of situations … [O]ne begins to manufacture linguistic entities which do not correspond to real things. … That’s where the River of Discernment runs into the Fortress of Tautology, into the sewers. It’s become invisible, but at the surface it can smell pretty bad.”


How I need a drink, alcoholic of course, after the tough chapters involving quantum mechanics!

That sentence is often offered as a mnemonic for pi — if we count the letters in each word we get 3.14159265358979. But systems like this are a bit treacherous: The mnemonic presents a memorable idea, but that’s of no value unless you can always recall exactly the right words to express it.

In 1996 Princeton mathematician John Horton Conway suggested that a better way is to focus on the sound and rhythm of the spoken digits themselves, arranging them into groups based on “rhymes” and “alliteration”:

                        _     _   _
            3 point  1415  9265  35
                     ^ ^
             _ _  _ _    _ _   __
            8979  3238  4626  4338   3279
              **  **^^          ^^   ****
             .   _    _   __   _    _      _ . _ .
       502 884  197 169  399 375  105 820  974 944
        ^  ^                       ^  ^
                59230 78164
                 _     _    _    _
              0628  6208  998  6280
               ^^   ^^         ^^
             .. _  .._
             34825 34211 70679
                         ^  ^

He walks through the first 100 digits here.

“I have often maintained that any person of normal intelligence can memorize 50 places in half-an-hour, and often been challenged by people who think THEY won’t be able to, and have then promptly proved them wrong,” he writes. “On such occasions, they are usually easily persuaded to go on up to 100 places in the next half-hour.”

“Anyone who does this should note that the initial process of ‘getting them in’ is quite easy; but that the digits won’t then ‘stick’ for a long time unless one recites them a dozen or more times in the first day, half-a-dozen times per day thereafter for about a week, a few times a week for the next month or so, and every now and then thereafter.” But then, with the occasional brushing up, you’ll know pi to 100 places!