Symmetries of p-polygons
Rolf Haag
TL;DR
The paper develops a symmetry-based classification of $p$-polygons with prime $p\ge5$ by partitioning equivalence-classes under rotations and distinguishing mirror images. It builds a rigorous framework leveraging the dihedral group and Burnside's Lemma, yielding closed-form counts for the total number of $p$-polygons, the regular subset, and the subsets with exactly one axis or no axis of symmetry: $|P(p)|=\dfrac{(p-1)!+(p-1)^2}{2p}$, $|X_p(p)|=\dfrac{p-1}{2}$, $|X_{1+}(p)|=\dfrac{p-1}{2}2^{\frac{p-3}{2}}\left(\frac{p-3}{2}\right)!$, $|X_1(p)|=\dfrac{p-1}{2}\left[2^{\frac{p-3}{2}}\left(\frac{p-3}{2}\right)!-1\right]$, and $|X_0(p)|=\dfrac{(p-1)^2+(p-1)!-p(p-1)2^{\frac{p-3}{2}}\left(\frac{p-3}{2}\right)!}{2p}$. The proofs adapt Golomb–Welch formulas to prime $p$ and provide constructive counting arguments for symmetry-bearing cases, with explicit representatives for $p=5$ and $p=7$ to illustrate the classifications. The results illuminate how symmetry degrees structure polygonal configurations and offer a concrete reference for enumerating distinct polygonal shapes under rotational equivalence. The work bridges combinatorial enumeration with geometric symmetry, providing exact counts and visual representatives that can inform related studies in polygon theory and symmetry classifications.
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.