Geometry and Number Theory on Clovers

Jerry Shurman
Department of Mathematics, Reed College

Abstract: Gauss determined the values of n for which the circle can be divided into n arcs of equal length by ruler and compass. His argument, using algebraic structure to solve a geometric problem, is the first proto-example of Galois theory. Abel solved the division problem for the lemniscate, and his method contains the beginnings of complex multiplication and even class field theory. Pierpont solved the circle problem with origami in place of ruler and compass.

This talk places these results in the general setting of m-clovers for all positive integers m, including the cardioid when m=1, the circle when m=2, the lemniscate when m=4, and a three-leaf clover when m=3 that apparently has not been studied until now. Analyzing origami division of the three-leaf clover and the lemniscate seems to require results from Galois theory, complex multiplication, and class field theory, in contrast to the elementary arguments by Gauss and Abel that led into these areas. The results of this talk are joint work with David Cox.