# Polyhedra with 8 Faces and 9 Vertices

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. In graph theory, the exterior of the polygon is considered the face. 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. Some polyhedra, for example those with two pentagon faces, are inherently ambiguous, because the net will be different depending on which pentagon is considered the base. Those cases can be sorted out by drawing auxiliary diagrams with both pentagons folded flat and regarded as the base. In a number of cases, equivalent polyhedra are shown, for the benefit of readers who may think they have discovered an unlisted case.

The original listing was compared with the output from the graph-listing program plantri.exe, and about 15 errors were found, either omissions or duplications. These have been corrected in the present listing. A program to draw nets given the plantri.exe face adjacencies was developed and each net was compared to the output. A few more duplications and omissions were found and corrected. Also, each net is shown in its topological variants. For example, the polyhedra 3333-4444 can have any one of the four quadrilateral faces as a base. Thus there can be up to four views of each polyhedron. There may be fewer if the polyhedron is symmetrical.

 Each of these polyhedra could have six different views, corresponding to each of the 4-gon faces. However, the topological distinctness of each is evident from the topology of the two triangular faces.

There are 21 (3333-44-55) polyhedra where the pentagons share an edge. If we fold the pair of pentagons flat and distort them, they form an octagon folded along a diameter. Each is shown below, with top and bottom views on the left and right, and the "unfolded" view in the center. The three cases in the second row are symmetrical; opposing sides are mirror images, and the top and bottom views are identical so there is only one view shown.

 The four (3333-44-55) cases shown at left have pentagons joined only at a vertex. The last example actually has two-fold rotation symmetry so both views are identical.