Prism product

The prism product is an operation that can be applied on any two polytopes. The prism product of two polytopes A and B is formed by placing a copy of A at each vertex of B, so that the hyperplanes of the copies of A intersect the hyperplane of B at a single point. Its facets will be the prism products of the facets of each polytope with the other polytope.

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

Prism products extend the notion of a prism, which results from the special case when one of the factors is a dyad. As such, they're 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+1 dyads.

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.

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.

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 simply correspond to prisms. The simplest 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 simply equal to the product of their volumes.

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

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

A duoprism made out of polytopes P and Q will have a symmetry order equal to the product of the symmetry orders of P and Q, except that the polytopes are congruent, in which case it will have double that symmetry order.

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.

Definition
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.

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.

Example
A cylinder is a prism-product of a line and a circle.

As such, one might represent a cylinder as a stack of coins, where each layer of height is repeated as a separate coin.

Alternately, one can make a cylinder out of a bundle of matches, each match represents the full height of the cylinder, and a point at the base.

The base of the cylinder is repeated for each point of height of the match, and the height is repeated for each point of the coin, by different matches.

The interior of the cylinder is created by the "interiors" of the match and coin. A match representing a line is interior except for the end-points. Likewise, the faces of the coin represent the interior of it. There is only two faces of coin showing in the cylinder, because all other points are at heights that represent interior of the match. Likewise, only those matches standing on the edge of the coin can be seen from outside.