We can create stars within polygons by picking a vertex and drawing a segment extending a certain consistent number of vertices from it. Most $2n$-gons seem to have one or two ways to make a star, e.g. the decagonal star, being segments connected every three vertices.
However, I am primarily focusing on odd-numbered polygons or $(2n+1)$-gons. It seems, that most odd polygons apart from the pentagon and triangle have at least two ways to start, one where each inner segment intersects at least three others, and a lesser one, where each segment overlaps at least two others.
Polygons with a prime number of vertices will often have the most variations, e.g. the $11$-gon with at least $4$ stars, spaced $5, 4, 3$, or $2$ vertices away from a given start vertex. Below are more $(2n+1)$-gons with the possible stars (I know of) inscribed:
If anyone can help give me a formula that will predict or explain the possible variations of inscribed stars for any n-gon, including the triangle and pentagon, please do so.