# Enneahedra with 9 vertices: faces 3333 44444

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay

These are Schlegel Nets; that is, one face (usually the one with the most edges) has been selected as a base and the polyhedron flattened into a plane within the enclosing polygon. To help with identifying faces, they are color-coded as follows:

• Triangles: White
• Pentagons: Green
• Hexagons: Light Blue
• Heptagons: Magenta
• Octagons: Red

Also, we are only concerned with topologically distinct polyhedra, that is, differing in number or type of faces and vertices. Thus, a triangular prism and a tetrahedron with one vertex truncated are topologically equivalent 5-hedra, a cube and rhombohedron are topologically equivalent 6-hedra, and so on.

General Notes on Polyhedron Enumeration

This class is very difficult to enumerate. With four triangle faces and five quadrilateral faces, there are no topologically unique faces to use as a base, and it is extremely hard to be sure that any given graph is really distinct and not just an isomorphism of some other graph. We can sort the graphs by the topology of the triangles. Classifying the topology of the triangles is akin to enumerating all the ways of connecting four points. There are eleven graphs with four vertices, shown in yellow below. Alternative representations are shown in red. However, triangles can either be joined at a vertex or an edge, leading to many more types. Triangle analogs to the point graphs are shown in green. There are 55 total polyhedra.

Since any of five quadrilateral faces can be chosen as a base, asymmetrical polyhedra generally have five different views. Those with mirror plane or two-fold rotation symmetry usually have three, and highly symmetrical polyhedra can have two. All disctinct views are shown for each polyhedron, but no attempt has been made to correlate the views of different polyhedra.

Many other triangle networks exist but only those shown in green actually occur among these polyhedra. The last two point graphs have not analogs here because these polyhedra only have 9 vertices. Four isolated triangles require 12 vertices and a pair plus two isolated triangles require 10 or 11. Ultimately, the only way to ensure a complete enumeration is to use the plantri.exe listing plus a program to draw each case.