Inspired by Solving for Xmas: how to make mathematical Christmas cards
36 points i around the shape each connected to points i +/- 12. On the circle, this results in all equilateral triangles.
72 points i around the shape each connected to point i * 2 (mod 72).
108 points i around the shape each connected to point i * 3 (mod 108).
90 points i around the shape each connected to point i ^ 2 (mod 90).
180 points i around the shape each connected to point i ^ 3 (mod 180).
18 points i around the shape each connected to point i ^ 4 (mod 18).