Log24

My feeling for the Greeks as mathematicians is every bit as inadequate as that for the youthful Gauss. I do not know whence came their curiosity and depth. Perhaps no-one does. We live in a highly structured environment dedicated to research. We earn our living by it and we pin our hopes of recognition on it, but the questions we ask and the problems we solve are determined more by tradition, more by our colleagues than by our own natural and spontaneous curiosity. We are seldom playful; our efforts are never simply for our own amusement. A brief romp with Greek mathematics in which we examine the construction of the pentagon at length may be an occasion to capture briefly the ludible spirit of the Greeks

An hour is also not enough for an adequate understanding of analytic geometric and complex numbers nor for a presentation of the algebra required for Gauss’s construction [of the 17-sided regular polygon]. The complex numbers are an enormously effective tool that swallows the geometry, but it will be good to ask ourselves how. Moreover the four-fold or sixteen-fold algebraic symmetry is far more subtle than the five-fold or seventeen-fold geometric symmetry. Since it will reappear again and in spades when, and if, we discuss Galois and Kummer, it is best to get used to it now.