Closo-undecaborate(11)-dianion-from-xtal-3D-bs-17

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an octadecahedron (or octakaidecahedron) is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

with 18

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

. No octadecahedron is

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

; hence, the name does not commonly refer to one specific polyhedron. In chemistry, "''the'' octadecahedron" commonly refers to a specific structure with C_2v symmetry, the

edge-contracted icosahedron In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices. Construction It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 ...

, formed from a regular

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...

with one edge contracted. It is the shape of the

closo In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by ...

-boranate ion [ B₁₁ H₁₁]²⁻.

Convex

There are 107,854,282,197,058 topologically distinct ''convex'' octadecahedra, excluding mirror images, having at least 11 vertices.Counting polyhedra
/ref> (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.)

Examples

The most familiar octadecahedra are the

heptadecagon In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. Regular heptadecagon A ''regular heptadecagon'' is represented by the Schläfli symbol . Construction As 17 is a Fermat prime, the regular heptadecagon is a c ...

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...

hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. Regular hexadecagon A ''regular hexadecagon'' is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbo ...

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

, and the

octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each ot ...

. The hexadecagonal prism and the octagonal antiprism are

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also f ...

, with

bases and

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

or equilateral triangular sides. Four more octadecahedra are also found among the

Johnson solid 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 Johns ...

s: the

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 de ...

, the

square orthobicupola 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 jo ...

, the

elongated square cupola In geometry, the elongated square cupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square cupola () by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron ...

(also known as the diminished rhombicuboctahedron), and the

sphenomegacorona In geometry, the sphenomegacorona is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. . Johnson uses the prefix ''spheno-'' t ...

. Four Johnson solids have octadecahedral duals: the

elongated triangular orthobicupola In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a triangular orthobicupola () by inserting a hexagonal prism between i ...

, the

elongated triangular gyrobicupola 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, w ...

, the gyroelongated triangular bicupola, and the

triangular hebesphenorotunda In geometry, the triangular hebesphenorotunda is one of the Johnson solids (). . It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic Plato's influence on Western culture was so pro ...

. In addition, some

uniform star polyhedra In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

are also octadecahedra:

References

{{Polyhedra Polyhedra