# SeventeenSides

While exchanging some techno-banter with an engineer/scientist/teacher recently, I suddenly remembered a couple of conversations with the late Caltech professor Jon Mathews back in 1975. Mathews was lecturing on mathematical methods of physics, and after a discussion of Fourier analysis I worked up my courage (as a nervous first-year grad student) to hang around after class and ask some questions. In proper pedagogic fashion, Mathews replied with further questions, and at one point I referred to "frequency space" --- a phrase which he challenged me to define.

Here's an attempt: Fourier transforms take a function (e.g., a sound such as musical chord) and analyze it in terms of sums of other functions (e.g., the pure tones that add up to make the original sound). It's kind of like sending a beam of light through a prism which breaks it up into its constituent colors --- or contrariwise, which takes a spectrum as input and merges it into a single beam. (Yep, a prism can do that.) My mental model of Fourier analysis associates it with a "real space" (including perhaps the graph of a signal versus time) and a "frequency space" (the results of the transform, totally equivalent in its information content). That's what I was trying to say when I spoke of "frequency space".

And remembering that brought to mind another classroom interaction I had with Professor Mathews. Warming up to perform a magical-mathematical feat of problem-solving (which Mathews was notorious for among the students) he suddenly asked, "Does anybody know what's on Gauss's tombstone?" I diffidently raised a hand and suggested, "A seventeen-sided figure?" based on some half-remembered biography of the great mathematician. "Right!" replied Mathews, happy to find that somebody else in the room was (1) awake, and (2) shared his interest in technical trivia.

Alas, we were both wrong on that one. In 1796, at age 19, Karl Friedrich Gauss did prove that a 17-sided regular polygon (aka heptadecagon, or heptakaidecagon) could be constructed with the classical tools of compass and straightedge. It was the first big step forward in that aspect of geometry since the ancient Greeks. But the figure doesn't decorate Gauss's gravestone, contrary to popular myth. "Popular" in some extremely limited circles, that is ...