Most Reinhardt polygons are sporadic

Abstract

A 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 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 $$r\ge 2$$ . 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.