![Ball-and-stick model of the octadecahedral closo-undecaborate ion, [B11H11] , as found in the crystal structure of the benzyltriethylammonium salt. Closo-undecaborate(11)-dianion-from-xtal-3D-bs-17.png](http://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Closo-undecaborate%2811%29-dianion-from-xtal-3D-bs-17.png/220px-Closo-undecaborate%2811%29-dianion-from-xtal-3D-bs-17.png)
In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.
In chemistry, "the octadecahedron" commonly refers to a specific structure with C2v symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [ B 11 H 11]2−.
There are 107,854,282,197,058 topologically distinct convex octadecahedra, excluding mirror images, having at least 11 vertices. [2] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
The most familiar octadecahedra are the heptadecagonal pyramid, hexadecagonal prism, and the octagonal antiprism. The hexadecagonal prism and the octagonal antiprism are uniform polyhedra, with regular bases and square or equilateral triangular sides. Four more octadecahedra are also found among the Johnson solids: the square gyrobicupola, the square orthobicupola, the elongated square cupola (also known as the diminished rhombicuboctahedron), and the sphenomegacorona. Four Johnson solids have octadecahedral duals: the elongated triangular orthobicupola, the elongated triangular gyrobicupola, the gyroelongated triangular bicupola, and the triangular hebesphenorotunda. Among near-miss Johnson solids, the chamfered cube is one notable non-uniform octadecahedron.
 Octagonal antiprism |  Square gyrobicupola |  Sphenomegacorona |  Elongated hexagonal bipyramid |  Chamfered cube |
In addition, some uniform star polyhedra are also octadecahedra:
 Octagrammic antiprism |  Octagrammic crossed-antiprism |  Small rhombihexahedron |  Small dodecahemidodecahedron |  Great rhombihexahedron |  Great dodecahemidodecahedron |
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance.
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, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a rotunda; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians Norman Johnson and Victor Zalgaller.
In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.
In geometry, the square cupola is the cupola with octagonal base. In the case of edges are equal in length, it is the Johnson solid, a convex polyhedron with faces are regular. It can be used to construct many polyhedrons, particularly in other Johnson solids.
In geometry, the elongated square gyrobicupola is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It was once mistakenly considered a rhombicuboctahedron by many mathematicians. It is not considered to be an Archimedean solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron. For this reason, it is also known as pseudo-rhombicuboctahedron, Miller solid, or Miller–Askinuze solid.
In geometry, the square orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola.
In geometry, the square gyrobicupola is one of the Johnson solids. Like the square orthobicupola, it can be obtained by joining two square cupolae along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.
In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola.
In geometry, the triangular orthobicupola is one of the Johnson solids. As the name suggests, it can be constructed by attaching two triangular cupolas along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron, twisted cuboctahedron or disheptahedron. It is also a canonical polyhedron.
In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in . It is an example of Johnson solid.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.
In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.
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 geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.
In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices.
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.