Duoprism

A duoprism is a class of polytopes formed as the Cartesian product of two polytopes. The simplest non-trivial duoprism is the triangular duoprism, which is the Cartesian product of two triangles. The dual of a duoprism is a duotegum. The duoprism is part of an infinite series of prism products, which include prisms, trioprisms and tetraprisms. The hypercubes are duoprisms made from lower-dimensional hypercubes.

If one of the polytopes is a point, then the resulting polytope is identical to the other polytope. If one of the polytopes is a line segment, then the resulting polytope is the prism of the other polytope. Neither of these cases are usually considered duoprisms.

A duoprism made out of polytopes a and b will have a symmetry order equal to the product of the symmetry orders of a and b, and duoprisms made out of two congruent polytopes will have double that symmetry order.

The vertex coordinates of a duoprism are given by all ordered pairs of the vertices of the two polytopes a and b. As such, they have a number of vertices equal to the product of the numbers of vertices of each polytope. Their facets will be the duoprisms based on one polytope and the facets of the other polytope.

The circumradius of a duoprism is equal to the square root of the sum of the squares of the circumradii of the polytopes that make it up.

The hypervolume of a duoprism is equal to the product of the hypervolumes of the two polytopes that "make up" the duoprism, just as the area of a rectangle is equal to the product of the lengths of its two orthogonal edges.

In 4D, the dichoral angle between two like cells of a duoprism is equal to the interior angle of the base polygon of the other kind of prism cell, and the dichoral angle between two dissimilar cells of the duoprism is 90°. For example, in the triangular-octagonal duoprism, the dichoral angle between octagonal prisms is 60°, the dichoral angle between triangular prisms is 135°, and the dichoral angle between triangular and octagonal prisms is 90°. In fact, this generalizes to other dimensions; all the angles are either identical to those of the bases or 90°.