The Star Polygon

A star polygon is an informal name given to polygons which, well... look like stars. A regular star polygon ⭐, by Keplar's definition, is a self-intersecting, equilateral, equiangular polygon.

Getting a little more technical, regular star polygons are denoted by a Schläfli symbol {p/q}, where p is the amount of vertices (pointy bits), and q is the density (the number of points to count over to connect one point to the next). To be a regular star polygon, p and q must not share any common factors (except 1, making them relatively prime). Also, q must be at least 2. If q is 1, you "just" get a normal old, regular polygon (like 🔷).

A regular star polygon can be drawn as a single continuous set of line segments, without picking your pencil off the paper. A non-regular star polygon (such as ✡️) requires two separate sets of line segments (see the two overlapping triangles?).

You can draw a regular star polygon by starting at any vertex of a regular, p-sided polygon. Then connect this vertex to another, non-adjacent vertex q vertices away. Continue this process until you get back to the starting point.

Play with the following sliders to see what different values for p and q look like. Not all values create a regular star polygon (p and q must be relatively prime 😃). Some of my favorite regular star polygons are {61,20}, {93,46}, {69,32}, and {7,2}.

Star polygons are found scattered throughout art and culture. For more information, check out Wikipedia's article on Star Polygons in Art and Culture and Star Polygons. A particularly interesting, star polygon-related rabbit hole that I suggest you check out is Penrose Tiling.