# Enneahedra with 10 vertices: 33 444444

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

This is an extraordinarily hard set to enumerate. The large number of quadrilateral faces means there can be up to 7 alternate base faces, and it is extremely hard to be sure that any view is unique, or merely a variation of another solid. The most effective strategy is to look at the topological relations of the two triangular faces. Views with a triangular face as a base sometimes result in the other triangle being impossibly tiny, however. Without plantri.exe and a graph-drawing program, it would be all but impossible to enumerate this class with confidence.

Because of the large number of quadrilateral faces, this class has an unusually large proportion of symmetrical polyhedra.

General Notes on Polyhedron Enumeration

## Asymmetrical Enneahedra

 Onefold (no) symmetry. Onefold (no) symmetry. Onefold (no) symmetry. The two triangles touch at a vertex. Onefold (no) symmetry. Two triangles share an edge. Onefold (no) symmetry. Two triangles share an edge. Onefold (no) symmetry.

## Two-Fold Rotational Symmetry

 This solid has two-fold symmetry so there are only four topologically distinct views. Twofold symmetry. Two-fold symmetry. Twofold symmetry.

## Mirror Plane Symmetry

 Mirror Plane Symmetry Mirror plane symmetry. Mirror plane symmetry. Mirror plane symmetry. Two triangles meet at an edge. Two triangles share an edge. Triangles are bisected by a mirror plane, so both triangles are topologically distinct .

## Two Mirror Planes

 A solid with two perpendicular mirror planes.