High symmetric n-polygons

TL;DR

This paper enumerates symmetric n-polygons with specific axes of symmetry and explores their relation to perfect numbers when n is a power of 2, contributing to combinatorial geometry and number theory.

Contribution

It provides a new enumeration of n-polygons with half the number of symmetry axes and establishes a connection to perfect numbers for powers of 2.

Findings

Enumeration of n-polygons with n/2 symmetry axes for even n

Relation between such polygons and perfect numbers when n is a power of 2

New insights into symmetry properties in polygon enumeration

Abstract

The present article includes the enumeration of $n$-polygons with a certain symmetry property: For an even number $n$ of vertices, we count the $n$-polygons with $\frac{n}{2}$ symmetry axes. In addition, if $n$ is a power of 2, we show the relation to the perfect numbers.