Dividing the circle

TL;DR

This paper analyzes approximate methods for constructing regular polygons, comparing Bion and Tempier techniques, to understand their errors and underlying reasons for their effectiveness in geometric constructions.

Contribution

It provides a mathematical comparison of Bion and Tempier approximation methods for polygon construction, explaining their accuracy and underlying principles.

Findings

Bion and Tempier methods have distinct error characteristics.

The paper explains why these approximation methods work.

Approximate constructions can be highly precise for certain polygons.

Abstract

There are known constructions for some regular polygons, usually inscribed in a circle, but not for all polygons - the Gauss-Wantzel Theorem states precisely which ones can be constructed. The constructions differ greatly from one polygon to the other. There are, however, general processes for determining the side of the $n$-gon (approximately, but sometimes with great precision), which we describe in this paper. We present a joint mathematical analysis of the so-called Bion and Tempier approximation methods, comparing the errors and trying to explain why these constructions would work at all.