![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.
 Octagonal antiprism |  Square gyrobicupola |  Sphenomegacorona |  Elongated hexagonal bipyramid |
In addition, some uniform star polyhedra are also octadecahedra:
 Octagrammic antiprism |  Octagrammic crossed-antiprism |  Small rhombihexahedron |  Small dodecahemidodecahedron |  Great rhombihexahedron |  Great dodecahemidodecahedron |
In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. The convex polyhedra with regular faces and symmetric vertices include also the five Platonic solids and the two infinite families of prisms and antiprisms; these are not counted as Archimedean solids. The pseudorhombicuboctahedron has regular faces, and vertices that are symmetric in a weaker sense; it is also not generally counted as an Archimedean solid. The Archimedean solids are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
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 is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ; it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a "Johnson solid".
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids. It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that map every vertex to every other vertex. It strongly resembles, but should not be mistaken for, the rhombicuboctahedron, which is an Archimedean solid. It is also a canonical polyhedron.
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 gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.
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 one of the Johnson solids. As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae. Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola.
In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.
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.
A pseudo-uniform polyhedron is a polyhedron which has regular polygons as faces and has the same vertex configuration at all vertices but is not vertex-transitive: it is not true that for any two vertices, there exists a symmetry of the polyhedron mapping the first isometrically onto the second. Thus, although all the vertices of a pseudo-uniform polyhedron appear the same, it is not isogonal. They are called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra.
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.