In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedra, each having a different pattern of vertex, edge, and face connections.Steven Dutch: How Many Polyhedra are There? None of them are regular.
Examples
Space-filling enneahedra
Slicing a rhombic dodecahedron in half through the long diagonals of four of its faces results in a self-dual enneahedron, the square diminished trapezohedron, with one large square face, four rhombus faces, and four isosceles triangle faces. Like the rhombic dodecahedron itself, this shape can be used to tessellate three-dimensional space.. An elongated form of this shape that still tiles space can be seen atop the rear side towers of the 12th-century Romanesque Basilica of Our Lady (Maastricht). The towers themselves, with their four pentagonal sides, four roof facets, and square base, form another space-filling enneahedron.
More generally, found at least 40 topologically distinct space-filling enneahedra.
Topologically distinct enneahedra There are 2606 topologically distinct convex enneahedra, excluding mirror images. These can be divided into subsets of 8, 74, 296, 633, 768, 558, 219, 50, with 7 to 14 vertices, respectively.Counting polyhedra A table of these numbers, together with a detailed description of the nine-vertex enneahedra, was first published in the 1870s by Thomas Kirkman.
References By the handshaking lemma, a face-regular polyhedron with an odd number of faces must have faces with an even number of edges, which for convex polyhedra can only be quadrilaterals. An enumeration of the dual graphs of quadrilateral-faced polyhedra is given by . Table 1, p. 619, shows that there is only one with nine faces.
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; see Definition 3.3, Figure 3.6, and related discussion
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External links Enumeration of Polyhedra by Steven Dutch