Octadecahedron

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.

Ball-and-stick model of the octadecahedral closo-undecaborate ion, [B₁₁H₁₁]²⁻, as found in the crystal structure of the benzyltriethylammonium salt.^[1]

In chemistry, "the octadecahedron" commonly refers to a specific structure with C_2v symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [B₁₁H₁₁]²⁻.

Convex

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.)

Examples

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

References

1. O. Volkov; W. Dirk; U. Englert; P. Paetzold (1999). "Undecaborates M₂[B₁₁H₁₁]: Facile Synthesis, Crystal Structure, and Reactions". Z. Anorg. Allg. Chem. 625 (7): 1193–1200. doi:10.1002/(SICI)1521-3749(199907)625:7<1193::AID-ZAAC1193>3.0.CO;2-L.
2. Counting polyhedra