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:

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.


All 9-hedra of this type require auxiliary views to ensure complete enumeration. These polyhedra have a vertex where four quadrilaterals meet. 

In the lower diagrams, the four meeting quadrilaterals are moved to the back of the polyhedron (hidden edges shown in green). Since those four quadrilaterals have 8 free edges, the remaining faces can be shown drawn on an octagonal net. Thus each polyhedron is shown twice, once in a square net and again in the octagonal net immediately below it.

All remaining 9-hedra of this type at least three quadrilaterals forming a strip. 

In the lower diagrams, the three quadrilaterals are moved to the back of the polyhedron (hidden edges shown in green). Since those three quadrilaterals have 8 free edges, the remaining faces can be shown drawn on an octagonal net. Thus each polyhedron is shown twice, once in a square net and again in the octagonal net immediately below it.

We cannot have five quadrilaterals meeting at a vertex because such a configuration would require 11 vertices.

We also cannot have four quadrilaterals forming a strip because that configuration requires ten vertices.

We also cannot have a cluster of 3 quadrilaterals and another of two not joined edge to edge. A cluster of three quadrilaterals requires eight vertices and that would leave only one vertex for the remaining pair of quadrilaterals.


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Created 10 June 1998, Last Update 7 June 1999

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