  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.
