Self-intersection

A polytope is self-intersecting if it has at least two distinct elements with interiors that intersect. For the sake of this definition, a vertex's location is considered to be its interior.

Self-intersecting polytopes cannot be convex. In Euclidean space, non-orientable polytopes must be self-intersecting.

Self-intersection often creates confusion for novice polytopists as it defies visual intuition. Historically, the Kepler-Poinsot solids were only studied and characterized millennia after the Platonic solids. However, it's usually more mathematically "natural" to restrict discussion to either convex polytopes or general polytopes permitting self-intersection, rather than the intermediate category of general polytopes excluding self-intersection. That said, some polytopists have specifically sought out non-convex polytopes without self-intersection, such as in Stewart toroids and some discussions of acrohedra.