Langton’s Ant

Set an ant down on a grid of squares and ask it to follow two rules:

  1. If you find yourself on a white square, turn 90° right, change the color of the square to black, and move forward one unit.
  2. If you find yourself on a black square, turn 90° left, change the color of the square to white, and move forward one unit.

That’s it. At first the ant will seem to mill around uncertainly, as above, producing an irregular jumble of black and white squares. But after about 10,000 steps it will start to build a “highway,” following a repeating loop of 104 steps that unfolds forever (below). Computer scientist Chris Langton discovered the phenomenon in 1986.

Will this happen even if some of the starting squares are black? So far the answer appears to be yes — in every initial configuration that’s been tested, the ant eventually produces a highway. If there’s an exception, no one has found it yet.

Asked and Answered

A college professor once offered the following creative final exam: Write a suitable final exam for this course and supply a key. The first paper handed in read ‘Final Exam: Write suitable final exam for this course and supply a key. Key: Any reasonable variation of the previous sentence = 100%.’

— Michael Stueben, Twenty Years Before the Blackboard, 1998

Image: Wikimedia Commons

In 1960 Jane Goodall watched a chimpanzee repeatedly poking pieces of grass into a termite mound in order to “fish” for insects, the first observation of tool use among animals. When she notified anthropologist Louis Leakey of her discovery, he responded with a telegram:


Night Crossing

In late March 1938, Antonio Carrelli received a letter and a telegram in short succession. Both were from Ettore Majorana, the brilliant Italian physicist who had recently joined the faculty of the Naples Physics Institute, where Carrelli was director.

The letter read, “Dear Carrelli, I made a decision that has become unavoidable. There isn’t a bit of selfishness in it, but I realize what trouble my sudden disappearance will cause you and the students. For this as well, I beg your forgiveness, but especially for betraying the trust, the sincere friendship and the sympathy you gave me over the past months. I ask you to remind me to all those I learned to know and appreciate in your Institute, especially Sciuti: I will keep a fond memory of them all at least until 11 pm tonight, possibly later too. E. Majorana.”

The telegram had been sent immediately afterward: “Dear Carrelli, I hope you got my telegram and my letter at the same time. The sea rejected me and I’ll be back tomorrow at the Hotel Bologna traveling perhaps with this letter. However, I have the intention of giving up teaching. Don’t think I’m like an Ibsen heroine, because the case is different. I’m at your disposal for further details. E. Majorana.”

On investigation it was found that Majorana had withdrawn all the money from his bank account and taken the night boat from Naples to Palermo on March 23. He had sent both messages from Palermo and then boarded the steamer to return to Naples on the night of March 25.

But there the trail ended. On the return journey Majorana had shared a compartment with a local university professor, but beyond this point no trace of him could be found. His family offered a reward of 30,000 lire for his whereabouts, and Enrico Fermi implored Mussolini for aid, citing the “deep brilliance” of Majorana’s physics, which he compared to those of Galileo and Newton. A police search found no body but offered no clues.

What happened to him? Theories abound: The most natural explanation, that he committed suicide, is discounted by both his family and the bishop of Trapani, citing his strong Catholic faith. (Also, it doesn’t explain the withdrawal of the money.) Other theories contend that he was murdered, that he fled physics because he foresaw the advent of nuclear weapons, that he had a spiritual crisis and joined a monastery, that he became a beggar, and that he moved to South America. No one knows.

(Barry R. Holstein, “The Mysterious Disappearance of Ettore Majorana,” from the Carolina International Symposium on Neutrino Physics, 2008.)

The Road Coloring Problem

Every road in this little town is a one-way street, and each street is colored either red or blue. This has a helpful effect: If you start at any house in town and follow the sequence blue-red-red three times in a row, you’ll always arrive at the yellow house.

If you follow blue-blue-red three times, you’ll always arrive at the green one.

In 1970 Roy Adler and Benjamin Weiss asked whether it’s always possible to create such a coloring in a given network; in 2009 Avraham Trahtman proved that, within certain constraints, it is.

Pi Without Circles

The sum of the squares of the reciprocals of the positive integers is π2/6.

The sum of their fourth powers is π4/90.

The sum of their sixth powers is π6/945.

The area of the region under the Gaussian curve y = ex2 is the square root of π.

The probability that two integers chosen at random will have no prime factor in common is 6/π2.

The integer 8 can be written as the sum of two squares of integers, m2 + n2, in four ways, when (m, n) is (2, 2), (2, -2), (-2, 2), or (-2, -2). The integer 7 can’t be written at all as the sum of such squares. Over a very large collection of integers from 1 to n, the average number of ways an integer can be written as the sum of two squares approaches π. Why?

In a Word

n. a hoarder of books

In the rare book collection of the archives at Caltech is a copy of Adrien-Marie Legendre’s 1808 text on number theory. It comes from the collection of Eric Temple Bell, who taught mathematics at Caltech from 1926 to 1953. Inside the book is an inscription in Bell’s handwriting:

This book survived the San Francisco Earthquake and Fire of 18 April, 1906. It was buried with about 600 others, in a vacant lot, before the fire reached the spot. The house next door to the lot fell upon the cache; the tar from the roof baked the 4 feet of dirt, covering the books, to brick, and incinerated all but 4 books, of which this is one. Signed: E. T. Bell. Book buried just below Grace Church, at California and Stockton Streets. House number 729 California Street.

During the Great Fire of London in 1666, Samuel Pepys came upon Sir William Batten burying his wine in a pit in his garden. Pepys “took the opportunity of laying all the papers of my office that I could not otherwise dispose of” and later buried “my Parmazan cheese, as well as my wine and some other things.” I don’t know whether he ever recovered them.

Moments of Inspiration

James Watt perfects the steam engine, 1765:

I had gone to take a walk on a fine Sunday afternoon. I had entered the Green and had passed the old washing house. I was thinking up on the engine at the time and had got as far as the herd’s house, when the idea came into my mind that as steam was an elastic body it would rush into a vacuum, and that if a communication were made between the cylinder and an exhausted vessel it would rush into it and might there be condensed without cooling the cylinder. I had not walked farther than the golf house when the whole thing was arranged clearly in my mind.

Charles Darwin realizes why species diverge, 1840s:

I can remember the very spot in the road, whilst in my carriage, when to my joy the solution occurred to me; and this was long after I had come to Down. The solution, as I believe, is that the modified offspring of all dominant and increasing forms tend to become adapted to many and highly diversified places in the economy of nature.

Henri Poincaré discovers the relation between automorphic functions and non-Euclidean geometries, 1881:

Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the result at my leisure.

Walter Cannon recognizes the fight-or-flight response, 1911:

As a matter of routine I have long trusted unconscious processes to serve me. … [One] example I may cite was the interpretation of the significance of bodily changes which occur in great emotional excitement, such as fear and rage. These changes — the more rapid pulse, the deeper breathing, the increase in sugar in the blood, the secretion from the adrenal glands — were very diverse and seemed unrelated. Then, one wakeful night, after a considerable collection of these changes had been disclosed, the idea flashed through my mind that they could be nicely integrated if conceived as bodily preparations for supreme effort in flight or in fighting.

William Rowan Hamilton conceives the fundamental formula for quaternions, 1843:

But on the 16th day of the same month — which happened to be a Monday, and a Council day of the Royal Irish Academy — I was walking in to attend and preside, and your mother was walking with me, along the Royal Canal, to which she had perhaps driven; and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery.

Hamilton adds: “Nor could I resist the impulse — unphilosophical as it may have been — to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely,

i2 = j2 = k2 = ijk = -1

which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.” The bridge now bears a permanent plaque marking Hamilton’s achievement (below), and mathematicians undertake an annual walk from Dunsink Observatory to commemorate it.
Image: Wikimedia Commons