Partition of Non-Regular Prime Stellated Cyclic Polygons into Rotation Classes
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
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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.
Thus there are $2$ equivalence classes, each with $5$ elements.
Matt Westwood suggests that these equivalence classes could be nicknamed fish and bat.
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-3}$ Wilson's Theorem: Theorem $\text {3-5}$