Got to thinking about this diagram so I did up this sketch. Noticed that the inside looked like a regular octagon: all the sides are the same length. But to be regular all the angles would have to the same also, and while they all look sort of the same, a little study of the external triangles convinced me that, no, the angles are not the same and so while the inner figure is an octagon, you can't really call it regular.
All of the lines in the figure are connected: you can draw the entire figure by starting at any vertex and drawing one line after another without lifting your pencil from the paper. This surprised me because the star pattern is one of the first things I learned about working on cars. Most cars have (used to have?) five lug nuts holding the wheels on. To avoid warping the wheel you followed a star pattern when tightening the lug nuts. This was easy with a five pointed star, you could start anywhere and then you just followed the lines as if you were drawing a five pointed star.
This doesn't work with a wheel with four or six lug nuts. You either go in a circle, or you use two separate sequences of skipping one bolt. So I'm surprised you can reach all points of an octagon by using a star pattern. Guess I never dealt with eight bolt wheels often enough for them to make an impression.