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.

All uniform polyhedra and Johnson solids are acrohedra, and thus cover many acrons. However, there are many other possible acrons that are not found in these polyhedra, and the existence of an acrohedron containing a given acron is a nontrivial question. If an acrohedra do exist, we can also ask which ones are as small as possible by some reasonable measure such as face count.

If we restrict our discussion to acrons containing exactly three convex polygons found in the non-prismatic uniform and Johnson solids (3, 4, 5, 6, 8, and 10) and whose respective internal angles sum to less than 360, there are 34 possible acrons. Most are covered by uniform polyhedra and Johnson solids. Specialized constructions are known for 5-4-3 (m* and G3), 6-5-4 (due to Klitzing and Conway), and 10-6-3 (Conway). No acrohedra are known for 5-5-4, 6-5-3, 6-5-5, 8-5-3, 8-5-4, 8-5-5, 8-6-3, or 10-8-3.

The smallest convex regular polygon not found in the non-prismatic uniform or Johnson solids is the heptagon. Acrons containing heptagons are particularly difficult to crack, and it is likely some are unsolvable, but 7-4-4 (heptagonal prism), 7-4-3, 7-6-4 (both McNeill), and 7-7-3 (small supersemicupola) have known acrohedra. The 7-4-3 construction generalizes to 9-4-3 and 11-4-3, in a family known as the pairwise augmented cupolae.