Acrohedron

An X-Y-Z acrohedron is a polyhedron with only regular faces where at least one vertex is surrounded by an X-gon, a Y-gon, a Z-gon, and no other faces. Such a vertex is called an X-Y-Z acron. For example, the triangular cupola is a 6-4-3 acrohedron. This definition readily generalizes to lists of four or more regular polygons, in which case the order must be properly reflected. For example, the icosidodecahedron is a 5-3-5-3 acrohedron, but not a 5-5-3-3 acrohedron (of which a valid example would be the pentagonal orthobirotunda). Polyhedra with coplanar faces that share an edge are not considered valid.

Given an acron, the existence of an acrohedron containing that acron is a nontrivial question. Some acrons are covered by known uniform polyhedra and Johnson solids. Others require specialized constructions; a 7-7-3 acrohedron was found by Mason Green and deemed the small supersemicupola. Some acrons have no known acrohedra, such as 5-5-4 and 6-5-3.