Symmetries of p-polygons

TL;DR

This paper classifies and counts equivalence-classes of p-polygons based on their symmetry properties, including regular, with one axis, and asymmetrical polygons, for p greater than 3.

Contribution

It provides a detailed enumeration of p-polygons with various symmetry properties, including explicit representatives for p=5 and p=7.

Findings

Counted equivalence-classes of regular polygons with p axes

Counted classes of polygons with exactly one axis of symmetry

Identified classes of asymmetrical p-polygons for specific p values

Abstract

In addition to general considerations, the present work includes the enumeration of the equivalence-classes of p-polygons with p vertices for p bigger than 3 with certain symmetry properties: 1. We count the equivalence-classes of p-polygons with p symmetry axes, the so called regular polygons. 2. We count the equivalence-classes of p-polygons with exactly one axis of symmetry. 3. We count the equivalence-classes of p-polygons with no axis of symmetry, the so called asymmetrical p-polygons. For p = 5 and p = 7 we show in all three cases a set of representatifs of the equivalenceclasses.