The eminent geometer H.S.M. Coxeter (not to be confused with a Gilbert and Sullivan operetta) gave a colloquium at Caltech on Feb. 8, 1977. I attended it, and while it was fresh in my mind, I wrote the following account. Coxeter came into the lecture hall amid a bunch of students. They had probably just had a departmental tea. The students were grungier than I expected. At Johns Hopkins, male students were expected to wear a tie to colloquia.
Coxeter looked at the usual empty seats in the front rows, and invited anyone to come closer and get a better view of the blackboard. He was bald on top, fringed with cute curly white hair, not large in stature, and continually grinned in an engaging way. As he paused, no one came forward. He repeated the invitation. One student got up and moved to the front, and the rest all applauded.
He spoke about certain patterns of numbers and their geometric relationships. It was all very easy to follow; he’s a good teacher, as I heard some of the students comment after the lecture. He did absent-mindedly make a few mistakes. He caught some of them. I never correct anyone else’s slip of the chalk, because it doesn’t impede my own understanding, and if it did impede another listener’s, it was up to that other to ask. Nit-picking just slows down the lecture.
At one point he had us all laughing. There is a number that crops up in innumerable ways. It’s φ=(1 +√5)/2, about 1.618. It’s called the golden section, mainly because if you take a rectangle whose sides are φ and 1, and cut off a 1×1 square, the remaining piece is the same shape as the original rectangle. That is, (φ-1):1 :: 1:φ.
Φ appears again in the Fibonacci series-the numbers 1, 1, 2, 3, 5, 8, 13, …, where each number is the sum of the previous two. The Fibonacci series developed from the question, if one pair of rabbits produces another pair of (baby) rabbits every month, and baby rabbits are ready to breed in their turn after two months have gone by, how many pairs of rabbits will there be after so many months? But Fibonacci numbers crop up in all sorts of unexpected ways. The number of seeds in one spiral of a sunflower or the number of spines around a pineapple is almost always a Fibonacci number, for instance. Now, it turns out that the ratio between consecutive Fibonacci (“fee-bone-otchy”) numbers approaches φ. The first few ratios are 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, …, which as decimal numbers are 1, 2, 1.5, 1.66666…, 1.6, 1.625, …; the farther on you go, the closer you get to 1.618.
In his talk, Coxeter alluded to the golden section, which is the length of a diagonal of a pentagon whose side is 1. He called it τ (the Greek lower-case tau), even though most people call it φ (the Greek letter phi). I pronounce φ “fie,” but many equally erudite people pronounce it “fee”). The Greek letters ξπφχψ are xi, pi, phi, chi, and psi. I aim for consistency, and rhyme them all with “pie” for π, and I don’t hear anyone calling π “pee.” Coxeter explained that τ is for τομοσ, which is Greek for ‘cut.’ He joked, “People who call it φ are just making a feeble pun on Fibonacci,” and everyone laughed. I remember one other risible moment in a math colloquium. One problem in abstract algebra is the classification and enumeration of groups. The visiting professor strove mightily and concluded, “So there are exactly eleven groups of this type.” The students laughed, because they realized that it had been ages since anyone had mentioned a number lower than twenty in their classes or lectures. As a rule, math grad students don’t deal with specific numbers; they represent them with letters or other symbols. Besides, most classes of groups that we knew had infinite numbers of members: the cyclic groups of order n, for any positive integer n; similarly for the symmetric groups, or groups of permutations on n elements, again for any n.