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 the trapezohedron.

There are two contested definitions of the word "antiprism" in higher dimensions. 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 the first definition, the cubic-octahedral prismatoid can be called a cubic antiprism or an octahedral antiprism, and the hexadecachoron as a tetrahedral antiprism. The second definition refers to an alternated prism where the base is alternable, such as the omnisnub 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, as it is actually an alternated decagonal ditetragoltriate.

Unlike prisms, antiprisms in four dimensions or greater generally have no uniform realization, because there are too many variables for all edges to be rescaled to equal length. 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.