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.

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°.