Pyramid product

The pyramid product is an operation that can be applied on any two polytopes. The pyramid product two polytopes A and B is formed by placing them in non-intersecting hyperplanes and lacing them together. The resulting polytope can be called a duopyramid or a disphenoid. Its facets will be the pyramid products of the facets of a polytope times the other polytope.

Pyramid products extend the notion of a pyramid, which results from the special case when one of the factors is a point.

The rank of the pyramid product of two polytopes A and B is equal to the sum of the ranks of A and B plus one. As a result, pyramid products tend to be relatively high dimensional. The simplest duopyramid that can’t be represented as a simple pyramid is the 5D square disphenoid.

Pyramid products are particularly applicable to simplices. The pyramid product of an m-simplex and an n-simplex is an (m+n+1)-simplex.

Pyramid products are closely related to tegum products, but are differentiated mainly by the additional dimension required by pyramid products. While the dual of the tegum product of two polytopes is the prism product of their duals, the dual of the pyramid product of two polytopes is simply the pyramid product of the duals.

A pyramid product is isogonal if both bases are isogonal and congruent to each other, isotopic if both bases are isotopic and congruent to each other, or noble if both bases are noble and congruent to each other.

Like the prism product and tegum product, the pyramid product is both commutative and associative.

The volume of the pyramid product of an m-polytope A and an n-polytope B is equal to the m-volume of A, times the n-volume of B, times the distance between the hyperplanes, divided over m!n!. The 0-volume of a point is taken as 1.

The product was invented by Wendy Krieger.

Definition
When dealing with convex polytopes, pyramid products can be easily defined as the convex hull of the vertex set formed by two polytopes in non-intersecting hyperplanes. However, this definition does not generalize to non-convex shapes.

The most general way to define the pyramid product is abstractly. The pyramid product of two abstract polytopes defined by posets P and Q is the direct product of P and Q. In other words, this is the poset on P×Q with the relation such that
 * $$(p_1,q_1)\le(p_2,q_2)\text{ iff }p_1\le p_2\text{ and }q_1\le q_2.$$

To make this concrete, it suffices to map all elements of the form (vertex, null) to the corresponding concrete vertices of P, and all elements of the form (null, vertex) to the corresponding concrete vertices of Q.

One can also define the pyramid product for any two point sets, by placing them in non-intersecting hyperplanes and taking the set of all convex combinations between points of both sets. However, this can't easily be turned into a working definition for polytopes, given that both constructing a general polytope's interior and a polytope from its interior are not always possible.