The equilateral small octagon of maximal width

TL;DR

This paper determines the maximal width of an equilateral small octagon, the first open case, and introduces a family of polygons approaching this maximum as the number of vertices increases.

Contribution

It solves the open problem of maximal width for equilateral small octagons and proposes a family of polygons nearing this maximum for larger even numbers of vertices.

Findings

Maximal width of equilateral small octagon is approximately 3.24% larger than regular octagon.

Introduces a family of equilateral small n-gons with widths within O(1/n^4) of the maximum.

Provides the first solution to the open problem for n=8.

Abstract

A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24\%$ larger than the width of the regular octagon: $\cos(\pi/8)$. In addition, the paper proposes a family of equilateral small $n$-gons, for $n=2^s$ with $s\ge 4$, whose widths are within $O(1/n^4)$ of the maximal width.