Aaron Jehle: Construction of the Regular Heptagon

Abstract

The constructability of numbers with ruler and compass was a problem already concerning the ancient Greeks and finally solved in modern algebra using Galois theory. We now know for certain that constructions like dividing an arbitrary angle into thirds, doubling the volume of a cube or constructing a regular heptagon are not possible using ruler and compass. With the so-called gardener's construction though—using two pins and a string—we can also construct ellipses and with these realize the aforementioned constructions. Using Galois Theory we will illustrate why this is possible by examining the fields of classically and conically constructible numbers and how to realize these constructions.