Prism product

The prism product, also known as the Cartesian product, is an operation that can be applied on any two polytopes. They extend the notion of a prism, which may be seen as a polytope (traditionally a polygon) extended through a line, to that of a polytope extended through any other. The simplest examples other than the prisms themselves are the polygonal duoprisms in 4D.

Polytopes resulting from n prism products can be called n-prisms. In the general case, these are called multiprisms or simply prismatic polytopes.

The special case of a prism results from whenever one of the factors is a dyad. As such, multiprisms are particularly applicable to hypercubes. The prism product of an m-hypercube and an n-hypercube is an (m + n)-hypercube. In particular, an n-hypercube can be seen as the prism product of n dyads.

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. Almost all research on uniforms thus focuses on finding the remaining uniforms.

Prism products are closely related to tegum products, given that the dual of the prism product of two polytopes is equal to the tegum product of their duals. They're also less obviously related to pyramid products, as abstractly, the prism product of two polytopes is a subset of their pyramid product. Like both other products, the prism product is both commutative and associative.

Like all other polytope products, the prism product has unique factorization. The neutral element of this product is the point.

Examples
A triangular prism (pictured to the right) is a prism product of a line segment and a triangle. As such, one can think of a triangular prism as either line segments wrapped around a triangle, or two triangles at each end of a line segment.

The interior of the triangular prism is created by the interiors of the line segment and triangle. For each point in the interior of the line segment, and each point in the interior of the triangle, there's a corresponding point inside the triangular prism at the height of the point in the line segment, over the corresponding point in the triangular base.

A more complicated example is the square-hexagonal duoprism (pictured above). It's the 4D prism product of a square and a hexagon. As such, one can think of it as either six squares wrapped around in a hexagon, or four hexagons wrapped around in a square.

The interior of the square-hexagonal duoprism is created by the interiors of the square and the hexagon. For each point in the interior of the square, and each point in the interior of the hexagon, there's a corresponding point inside the square-hexagonal duoprism whose projections into a square face or into a hexagonal face are the points we chose.

Elements
The elements of a prism product are the prism products of their elements. This observation is in fact the motivation for the formal definition for abstract polytopes – see below.

To get the element counts of a duoprism, one can write polynomials P(x) and Q(x) for its factors, so that the coefficient ak of P(x) equals the number of elements of rank k on the first polytope, and ditto for Q(x) and the second polytope. The element counts can then be read off from the polynomial P(x)Q(x). Contrast this with the pyramid product, where the single element of minimal rank is included as well.

Properties
The rank of the prism product of two polytopes A and B is equal to the sum of their ranks. As a result, up to 3D, prism products correspond to prisms. The simplest non-degenerate duoprism that can't be represented as a prism is the 4D triangular duoprism.

Any duoprism based on two uniform, scaliform, isogonal, or CRF polytopes will also maintain those properties. In addition, the duoprism of 2 identical isotopic polytopes is also isotopic.

The volume of the prism product of a set of polytopes is equal to the product of their volumes. This generalizes the fact that the volume of a prism equals the volume of its base times its height.

The circumradius of the prism product of a set of polytopes equals the square root of the sum of the squared circumradii of each of the polytopes.

The vertex figure of a duoprism is the pyramid product of the vertex figures of the components.

Symmetries
A duoprism made out of polytopes P and Q with symmetry groups G and H will, most of the time, have a symmetry group equal to the direct product G × H. There's some exceptions to this rule, however. If P and Q are congruent, there will also be a symmetry swapping the bases, making the symmetry group at least the wreath product G ≀ S2. In other exceptional cases, the symmetry group might be even larger, as is the case with prism products of hypercubes.

Definition
Intuitively, the prism product of two polytopes A and B can be formed by placing a copy of A at each vertex of B, so that the the copies of A lie in orthogonal subspaces to the subspace of B. Its facets will be the prism products of the facets of each polytope with the other polytope.

There's two ways to formalize this intuition, depending on what mathematical objects one is talking about.

Convex polytopes
The prism product of two convex polytopes corresponds precisely to their Cartesian product as sets. Recall that the cartesian product A × B is the set of all pairs (a, b) where a ∈ A and b ∈ B. One identifies a pair of a point with m coordinates and a point with n coordinates as a point with m + n coordinates. One can then prove that this set of points forms a convex polytope.

Abstract polytopes
The most general way to define the prism product is abstractly. The prism product of two polytopes defined by posets P and Q is the direct product of P and Q, with all of the elements (p, q) where exactly one of p and q is of minimal rank taken out. In other words, this is the poset on
 * $$\{(p, q):p\in P\text{ and }q\in Q\text{ and either none or both of }p\text{ and }q\text{ are minimal}\},$$

with the relation such that
 * $$(p_1,q_1)\le(p_2,q_2)\text{ iff }p_1\le p_2\text{ and }q_1\le q_2.$$

This can be contrasted with the pyramid product, which doesn't omit any elements from the direct product, or with the tegum product, which omits those with elements of maximal rank instead.

To make this concrete, it suffices to map all elements of the form (vertex, vertex) to the points resulting from concatenating the coordinates of the corresponding concrete vertices.