Ruler and compass constructions of the equilateral triangle and pentagon in the lemniscate curve.

Abstract

A classical theorem of Abel on regular N-gons in the lemniscate curve mimics the analogous result of Gauss for the circle. In both cases, one gets the same criterion: the regular N-gons constructible with ruler and compass are those with number of sides N = 2^v· p_1 · · · p_r, where p_i are distinct Fermat primes. The hands-on drawing of such regular polygons with ruler and compass in the circle case is fairly well-known (at least when N is small). Here we present the practical constructions for the regular triangle and pentagon in the lemniscate case.