The equilateral small octagon of maximal width

Abstract

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