Most Reinhardt polygons are sporadic

TL;DR

This paper investigates Reinhardt polygons, revealing that sporadic types are more common than periodic ones for most sizes, especially starting at n=105, and provides formulas for counting sporadic cases.

Contribution

It demonstrates that sporadic Reinhardt polygons outnumber periodic ones for almost all n, and derives a counting formula for n=2pq cases.

Findings

Sporadic Reinhardt polygons are more numerous than periodic ones for most n.

The first n where sporadic polygons dominate is 105.

A formula for counting sporadic polygons when n=2pq is established.

Abstract

A \textit{Reinhardt polygon} is a convex $n$-gon that, for $n$ not a power of $2$, is optimal in three different geometric optimization problems, for example, it has maximal perimeter relative to its diameter. Some such polygons exhibit a particular periodic structure; others are termed \textit{sporadic}. Prior work has described the periodic case completely, and has shown that sporadic Reinhardt polygons occur for all $n$ of the form $n=pqr$ with $p$ and $q$ distinct odd primes and $r\geq2$. We show that (dihedral equivalence classes of) sporadic Reinhardt polygons outnumber the periodic ones for almost all $n$, and find that this first occurs at $n=105$. We also determine a formula for the number of sporadic Reinhardt polygons when $n=2pq$ with $p$ and $q$ distinct odd primes.