In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. [1] None of them are regular.
In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. [1] None of them are regular.
The most familiar enneahedra are the octagonal pyramid and the heptagonal prism. The heptagonal prism is a uniform polyhedron, with two regular heptagon faces and seven square faces. The octagonal pyramid has eight isosceles triangular faces around a regular octagonal base. Two more enneahedra are also found among the Johnson solids: the elongated square pyramid and the elongated triangular bipyramid. The three-dimensional associahedron, with six pentagonal faces and three quadrilateral faces, is an enneahedron. Five Johnson solids have enneahedral duals: the triangular cupola, gyroelongated square pyramid, self-dual elongated square pyramid, triaugmented triangular prism (whose dual is the associahedron), and tridiminished icosahedron. Another enneahedron is the diminished trapezohedron with a square base, and 4 kite and 4 triangle faces.
 Heptagonal prism |  Elongated square pyramid |  Elongated triangular bipyramid |
 Dual of triangular cupola |  Dual of gyroelongated square pyramid |  Dual of tridiminished icosahedron |
 Square diminished trapezohedron |  Truncated triangular bipyramid, near-miss Johnson solid, and associahedron. |  Polyhedron with the skeleton of Herschel graph |
The Herschel graph represents the vertices and edges of the Herschel enneahedron above, with all of its faces quadrilaterals. It is the simplest polyhedron without a Hamiltonian cycle, [2] the only convex enneahedron in which all faces have the same number of edges, [3] and one of only three bipartite convex enneahedra. [4]
The smallest pair of isospectral polyhedral graphs are enneahedra with eight vertices each. [5]
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. [6] 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, Goldberg (1982) found at least 40 topologically distinct space-filling enneahedra. [7]
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. [8] A table of these numbers, together with a detailed description of the nine-vertex enneahedra, was first published in the 1870s by Thomas Kirkman. [9]
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
In geometry, an n-gonaltrapezohedron, n-trapezohedron, n-antidipyramid, n-antibipyramid, or n-deltohedron, is the dual polyhedron of an n-gonal antiprism. The 2n faces of an n-trapezohedron are congruent and symmetrically staggered; they are called twisted kites. With a higher symmetry, its 2n faces are kites.
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
In geometry, the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid.
In geometry, a trigonal trapezohedron is a polyhedron with six congruent quadrilateral faces, which may be scalene or rhomboid. The variety with rhombus-shaped faces faces is a rhombohedron. An alternative name for the same shape is the trigonal deltohedron.
In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.
In geometry, the great rhombihexahedron (or great rhombicube) is a nonconvex uniform polyhedron, indexed as U21. It has 18 faces (12 squares and 6 octagrams), 48 edges, and 24 vertices. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral.
In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces.
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph, and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs.
In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing each facet of P with a pyramid. In some cases, the pyramid is chosen to have regular sides, often producing a non-convex polytope; alternatively, by using sufficiently shallow pyramids, the results may remain convex. Kleetopes are named after Victor Klee, although the same concept was known under other names long before the work of Klee.
In geometry, an octadecahedron is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.
In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular n-gonal base face, n triangle faces around the base, and n kites meeting on top. The kites can also be replaced by rhombi with specific proportions.
A hexadecahedron is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism.
In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.