Number of Regular Stellated Odd n-gons

Theorem

Then there are $\dfrac {n - 1} 2$ distinct regular stellated $n$-gons.

Let the $n$ vertices of $P$ be $p_1, p_2, \dotsc, p_n$.

These will be arranged on the circumference of a circle $C$, dividing $C$ into $n$ arcs of equal length.

Once we have chosen the first side of $P$, the others are all the same length and are completely determined by that first side.

Without loss of generality, the first vertex of $P$ is chosen to be $p_1$ We can choose that first side as follows:

$p_1 p_2, p_1 n_3, \ldots, p_1 p_{n - 1}, p_1 p_n$

But we have that:

$\size {p_1 p_2} = \size {p_1 p_n}$

$\size {p_1 p_3} = \size {p_1 p_{n - 1} }$

and so on, down to:

$\size {p_1 p_{\paren {n - 1} / 2} }= \size {p_1 p_{\paren {n + 1} / 2} }$

where $\size {p_a p_b}$ denotes the length of the line $p_1 p_b$.

So for the $n - 1$ lines that are chosen for the first side of $P$, each is paired with another of the same length.

Hence there are $\dfrac {n - 1} 2$ ways of choosing the first side of $P$.

Thus there are $\dfrac {n - 1} 2$ distinct regular stellated $n$-gons.

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-3}$ Wilson's Theorem: Theorem $\text {3-5}$