Prism product

The prism product is an operation that can be applied on any two polytopes to obtain a new one. It generalizes the notion of a prism, which corresponds to the prism product of a polytope and a dyad. The prism product of two polytopes is called their duoprism.

When talking specifically about convex polytopes, if we identify each polytope with its interior, the prism product corresponds with the better-known Cartesian product. That is, the prism product of an m-dimensional polytope and an n-dimensional polytope is the set of all points in m + n dimensions whose first m coordinates correspond to a point on the interior on the first polytope, and whose last n coordinates correspond to a point on the interior of the other polytope. Nevertheless, while the Cartesian product is defined only for convex polytopes, the prism product is defined for non-convex and even skew polytopes.

The prism product is notable since the prism product of any two uniform polytopes with the same edge length is always uniform. This leads to the existence of a great many families of uniform polytopes in any given dimension, often grouped together as the prismatic uniforms. Most research on uniforms thus focuses on finding the remaining uniforms.

Up to symmetry, the prism product is both commutative and associative. Therefore, one can unambiguously define the prism product of any finite set of polytopes.

Trioprisms
A trioprism is the Cartesian product of three polytopes. The simplest non-trivial trioprism is the triangular trioprism, which is the Cartesian product of three triangles. The dual of a trioprism is a triotegum. Trioprisms can also be constructed as a duoprism of a polytope and another duoprismatic polytope. Trioprisms made out of two congruent polytopes will have double the symmetry order, and if all three constituent polytopes are congruent, it will have six times the symmetry order.

Like duoprisms, the vertex coordinates of a trioprism is determined by all ordered pairs of the three polytopes a, b, and c, and the hypervolume is equal to the product of the hypervolumes of the three constituent polytopes.

Multiprisms
An n-multiprism is the Cartesian product of n polytopes. A duoprism is a 2-multiprism, a trioprism is a 3-multiprism, a tetraprism is a 4-multiprism, a pentaprism is a 5-multiprism, etc. A 1-multiprism is any polytope as it is the product of only one polytope.