Partition of Non-Regular Prime Stellated Cyclic Polygons into Rotation Classes

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Theorem

Let $p$ be an odd prime.

Let $C$ be a circle whose center is $O$.

Consider the set $P$ of $p$ points on the circumference of $C$ dividing it into $p$ equal arcs.

Let $S$ be the set of all non-regular stellated $p$-gons whose vertices are the elements of $P$.

Let $\sim$ denote the equivalence relation on $S$ defined as:

$\forall \tuple {a, b} \in S \times S: a \sim b \iff$ there exists a plane rotation about $O$ transforming $a$ to $b$.


Then the $\sim$-equivalence classes of $S$ into which $S$ can thereby be partitioned all have cardinality $p$.


Proof


Examples

Pentagons

The equivalence classes by rotation of the non-regular stellated pentagons whose vertices are equally spaced on the circumference of a circle are depicted thus.


Stellated-Pentagons-Rotation-Classes.png


Thus there are $2$ equivalence classes, each with $5$ elements.


Matt Westwood suggests that these equivalence classes could be nicknamed fish and bat.


Sources