Antiprism (disambiguation)

An antiprism, defined in the three-dimensional sense, is a polyhedron that consists of two bases in opposite orientations connected by triangles. The dual of an antiprism is a trapezohedron.

Uniform antiprisms exist for any regular polygon {n/d} where n/d > 1.5. When n/d < 2, the triangular faces of the corresponding antiprism cross its rotational symmetry axis, and thus these antiprisms are often called retroprisms or crossed antiprisms.

Antiprisms make up one of the two infinite families of uniform polyhedra, the other being the family of prisms. These are in fact related, as the {n/d} antiprism can be constructed by alternation of a {2n/d}-gonal prism.

The edges of a uniform antiprism lie on a hyperboloid of one sheet. This is more apparent in star antiprisms where edges cross.

Coordinates
Coordinates for an {n/d} antiprism are given by
 * $$\left(\cos\left(\frac{k\pi}{n}\right),\ \sin\left(\frac{k\pi}{n}\right),\ (-1)^kh\right),$$

for k ranging from 0 to 2n – 1. If the antiprism is uniform, then h is specifically given by
 * $$h=\frac12\sqrt{\cos\left(\frac{\pi}{n}\right)-\cos\left(\frac{2\pi}{n}\right)}.$$

If furthermore the entire figure has unit edge length, the coordinates need to be scaled down by 2sin(π/n).

Extensions to higher dimensions
There are two contested definitions of the word "antiprism" in higher dimensions, along with one closely related concept.

Dual bases
The first definition refers to a polytope with vertices lying in two parallel planes, where the bases are duals to each other and are connected by various pyramidal facets. For example, under this definition, the octahedron atop cube segmentochoron can be seen as cubic antiprism or an octahedral antiprism, and the hexadecachoron can be seen a tetrahedral antiprism. Unless their bases are self-dual, as happens with the simplex cases, which produce cross polytopes, and with the 5D icositetrachoric antiprism, these antiprisms are generally not isogonal, and unless their bases are regular, they are not even CRF. This is the only definition that can be applied to any polytope.

These antiprisms generally tend to have some form of lower antiprism as a vertex figure, generally based on the vertex figure of either base. For instance the octahedron atop cube case mentioned above has a square antiprism vertex figure at the octahedral layer.

Alternated prisms
The second definition refers to an alternated prism where the base is alternatable, such as the snub cubic antiprism, derived from the great rhombicuboctahedral prism, and the hexadecachoron (tetrahedral antiprism), derived from the tesseract (cubic prism). Unlike the first definition, it only includes bases that are congruent and does not necessarily imply self-duality.

Duoantiprisms, following from the second definition, are alternations of duoprisms. The grand antiprism is confusingly named as such, although it is actually an alternated decagonal ditetragoltriate.

Unlike prisms, antiprisms in this sense, as for arbitrary alternations, in four dimensions or greater generally have no uniform realization, because there are too many edge lengths to be rescaled to equal length. Exceptions do exist in any dimension, as the demihypercubes, derived from hypercubes, can always be made uniform.

The vertex figure of an antiprism (under the second definition) is a wedge derived from a simplex two dimensions lower than the antiprism and the base's vertex figure. For example, the vertex figure of a demipenteract (hexadecachoric antiprism) is a rectified pentachoron, which is a wedge of a tetrahedron and an octahedron.

Alterprisms
A closely related concept to antiprisms are alterprisms, constructed by lacing two geometrically-identical polytopes either in some gyrated orientations together (i.e. tutcup) or in the same orientation with intersecting laced facets (i.e siidcup). Examples include laces of polytopes with simplecial or demicubic symmetry in any dimension, laces with icositetrachoric symmetry, or laces with the symmetry of the E6 family. Many of these polytopes are scaliform. A siimple 4D example of this would be the truncated tetrahedral cupoliprism.

The only uniform 4D cases, known as the Johnson antiprisms, are the hexadecachoron xo3oo3ox&#x, the small ditrigonary icosidodecahedral antiprism xo5/2ox3oo*a&#x, the great ditrigonary icosidodecahedral antiprism xo5/4ox3oo*a&#x, and the blend of the latter two, the ditrigonary dodecadodecahedral antiprism.