The spiral of Theodorus, numerical analysis, and special functions

摘要

Theodorus of Cyrene (ca. 460–399 B.C.), teacher of Plato und Theaetetus, is known for his proof of the irrationality of n, n=2,3,5,…,17. He may have known also of a discrete spiral, today named after him, whose construction is based on the square roots of the numbers n=1,2,3,…. The subject of this lecture is the problem of interpolating this discrete, angular spiral by a smooth, if possible analytic, spiral. An interesting solution was proposed in 1993 by P.J. Davis, which is based on an infinite product. The computation of this product gives rise to problems of numerical analysis, in particular the summation of slowly convergent series, and the identification of the product raises questions regarding special functions. The former are solved by a method of integration, in particular Gaussian integration, the latter by means of Dawson’s integral und the Bose–Einstein distribution. Number-theoretic questions also loom behind this work.