Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

58 the mathematical infinityFigure 2.1. Archimedes’s approximations to the circle.length of the corresponding polygon circumscribed to the circle (Fig. 2.1). Startingwith a given regular polygon with n sides, there is a simple geometrical constructionfor doubling the number of sides to 2n. When n gets larger and larger, the two polygonsbecome closer and closer to the circle, and their perimeters yield better and betterapproximations to π. Archimedes used this method, starting with the hexagon and doublingthe sides four times to reach 96 edges, to prove rigorously that 3 1071