The equilateral small octagon of maximal width

2021/09/30 by Christian Bingane, Charles Audet · 1 voice
Mathematics · #math.MG #math.CO #math.OC

paper · pdf · doi:10.1090/mcom/3733

arxiv published 2021/09/30 · arxiv updated 2022/02/03

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 ≥ 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(π/8). In addition, the paper proposes a family of equilateral small n-gons, for n=2^s with s≥ 4, whose widths are within O(1/n⁴) of the maximal width.

Discussions

• could be a square or more generally a rhombus. Each such shape has a width. Now, here's the question: for a fixed n, what is the maximum width of all small equilateral n-gons? According to this articl [bsky, 0 points, 1 comments]

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