User:Tomruen/Prismatic polytope

Polytopes expressible as Cartesian products of lower-dimensional polytopes (of dimension 1 or greater). When all the lower-dimensional polytopes are uniform and have the same edge length, the resulting Cartesian product is a uniform prismatic polytope.

The Cartesian product of any polytope and a dyad is called a prism; the polytope is the base of the prism, and the facets joining the top and bottom bases are the lateral facets. The Cartesian product of two regular polygons of the same edge length is a uniform duoprism or double prism.

A hyperprism is a prism of more than three dimensions.

A very general kind of convex prism may be formed in n-space by constructing the convex hull of two (n–1)-dimensional polytopes in parallel hyperplanes.

In three dimensions, it is convenient to give some of the symmetric prisms special names:

• A symmetric prism is one that has a symmetric polygon for both bases. A prism all of whose lateral faces are rectangles or squares is an orthoprism, and it is usually the kind of prism that is meant by the term prism for a polyhedron.
• A prism all of whose lateral faces are triangles is an antiprism. This may be a very general kind of polyhedron, but the term is often restricted to the figure formed by two congruent, antialigned, parallel regular polygons connected by congruent isosceles triangles.
• If the vertex figures of an antiprism are crossed trapezoids, it becomes a retroprism.
• A prism whose bases are two isogonal even-sided polygons with unequal sides, situated so that the long sides of either are parallel to the short sides of the other, may have congruent trapezoids or neckties for its lateral faces. Such a prism is either a loxoprism (if the faces are trapezoids; loxo comes from Greek for “slanted”) or a retroloxoprism (if the faces are neckties).
• If a prism and an antiprism are based on identical polygons oriented the same way, and they have the same height, they can be blended into a toroprism: a prism that has only the two kinds of lateral faces, and holes (toro comes from Greek for “hole”) where the bases were. The holes may sometimes penetrate the prism from base to base.

Many kinds of symmetric prisms and antiprisms may be obtained by symmetrically faceting an orthoprism; all 25 different prisms, antiprisms, loxoprisms, and other combinations appear in this diagram [1].

[2]