Pyramid product

The pyramid product, also known as the join, is an operation that can be applied on any two polytopes. The pyramid product of two polytopes is formed by placing them in non-intersecting hyperplanes and lacing them together. The resulting polytope can be called a duopyramid or a disphenoid. (Note that "disphenoid" most commonly (exclusively in the professional literature) refers to a specific type of tetrahedron that is a special case of this operation (a tetragonal, rhombic, digonal or phyllic disphenoid).) Its facets will be the pyramid products of the facets of each polytope with the other polytope.

Polytopes resulting from n pyramid products can be called n-pyramids. In the general case, these are called multipyramids.

Pyramid products extend the notion of a pyramid, which results from the special case when one of the factors is a point. As such, they're particularly applicable to simplices. The pyramid product of an m-simplex and an n-simplex is an (m + n + 1)-simplex. In particular, an n-simplex can be seen as the pyramid product of its n + 1 vertices.

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

Properties
The rank of the pyramid product of two polytopes is equal to the sum of their ranks 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.

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. A pyramid product of regular-faced polytopes with circumradii $$r_1$$ and $$r_2$$ can be made with regular faces if and only if $$\sqrt{r_1^2+r_2^2} < 1$$.

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, times $$\frac{m!\,n!}{(m+n+1)!}$$. The 0-volume of a point is taken as 1.

To get the element counts of a duopyramid, 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+1 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).

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
 * $$P\times Q=\{(p,q):p\in P\text{ and }q\in Q\}$$

with the order 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.