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 the second in an infinite series of multiprisms, 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. Duoprisms made out of two congruent polytopes will have double the symmetry order.

The vertex coordinates of a duoprism is determined by all ordered pairs of the two polytopes a and b. As such, they have a number of vertices equal to the product of the number of vertices of each polytope.

The hypervolume of a duoprism is equal to the product of the hypervolumes of the two polytopes that "make up" the duoprism. This can explain why the area of a square (or rectangle) is equal to the product of the lengths of its sides, or why the volume of a cube (or rectangular prism) is equal to one face's area times the length of an edge not part of that face. For 4-dimensional duoprisms, which are "made up" of two polygons, the formula A=n/(4tan(π/n)) can be used to find the area of an n-sided polygon with unit edge length.

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 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°.