Classical Construction Problems

Abstract

The ancient Greeks searched for a way of using straightedge and compass to trisect an arbitrary angle, and to draw a segment of length $\sqrt[3]{2}$. They also tried to’ square the circle’, that is, construct a segment of length $\sqrt \pi$. Finally, they struggled to find straightedge and compass constructions for regular polygons with 7, 9, 11, 13, and 17 sides. In all this they failed, but it was not proved until the nineteenth century that the reason for their failure was that all these problems are insoluble — except one. In 1796 Gauss discovered a straightedge and compass construction for the regular 17-sided polygon. It was this discovery, the first advance on construction problems in 2000 years, that motivated Gauss to devote himself to mathematics.