Self-intersection
A polytope or polytope compound 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. (For example, if a unit cube has a square pyramid excavated from it so the apex of the pyramid touches a face of the cube, the resulting polyhedron is self-intersecting.)
Non-self-intersecting polytopes are sometimes called acoptic, and polygons that fit this definition are often called simple. In the context of regular and uniform polytopes, self-intersecting polytopes are also called star polytopes.
Self-intersection is dependent on the definition of "interior". Skew elements generally have no defined interior, so self-intersection is not generally defined for polytopes with skew elements[note 1]. For non-skew polytopes, "interior" is technically dependent on filling method, but if two distinct filling methods agree on the definition of interior for non-self-intersecting polytopes, induction shows that they will also agree on which polytopes are self-intersecting. As there is broad agreement on filling methods for non-self-intersecting polytopes, in practice self-intersection is virtually unambiguous for non-skew polytopes in Euclidean space.
Self-intersecting polytopes cannot be convex. In Euclidean space, non-orientable polytopes must be self-intersecting or skew.
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.
Notes[edit | edit source]
- ↑ Self-intersection can be defined for polytopes that are only skew at the highest level (e.g. the square duocomb or the skew cube), or for polytopes that are skew by virtue of their vertex figure (e.g. the mucube or the zigzag), since we only care about the interiors of proper elements.