Tegum product

The tegum product, also known as the direct sum or free join, is an operation that can be applied on any two polytopes. The tegum product of two polytopes A and B is formed by placing them in hyperplanes with a single point as intersection, and lacing them together. The resulting polytope can be called a duotegum. Its facets will be the pyramid products of the facets of a polytope times the other polytope.

The word "tegum" is derived from the Latin word "tegere", meaning "to cover". This alludes to how tegums "wrap about" the polytopes that define them.

Polytopes resulting from n tegum products can be called n-tegums. In the general case, these are called multitegums.

Tegum products extend the notion of a tegum (also sometimes called a bipyramid), which results from the special case when one of the factors is a dyad. As such, they're particularly applicable to orthoplices. The tegum product of an m-orthoplex and an n-orthoplex is an (m+n)-orthoplex. In particular, an n-orthoplex can be seen as the tegum product of n dyads.

Tegum products are closely related to pyramid products, but are differentiated mainly by the additional dimension required by pyramid products. They're also related to the prism product, given that the dual of the tegum product of two polytopes is the prism product of their duals. Like both of these products, the tegum product is both commutative and associative.

Properties
The rank of the tegum product of two polytopes is equal to the sum of their ranks. As a result, up to 3D, tegum products simply correspond to tegums. The simplest duotegum that can’t be represented as a tegum is the 4D triangular duotegum.

A duotegum is isogonal if both bases are isogonal and congruent to each other, isotopic if both bases are isotopic, or noble if both bases are noble and congruent to each other. In general, a duotegum can't be made to have only regular faces, except in special cases such as tegum, or in a few rare instances such as the pentagonal-pentagrammic duotegum. Specifically, the tegum product of regular-faced polytopes where both are at least 2-dimensional with circumradii $$r_1$$ and $$r_2$$ can be made with regular faces if and only if $$r_1^2+r_2^2 = 1$$.

To get the element counts of a duotegum, 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, excluding the single maximal element, 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 maximal elements are not omitted.

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

The most general way to define the tegum product is abstractly. The tegum product of two abstract 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 maximal rank taken out. In other words, this is the poset on
 * $$\{(p, q):p\in P\text{ and }q\in Q\text{ and either none or both of }p\text{ and }q\text{ are maximal}\},$$

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 prism product, which omits those with elements of minimal rank instead. 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 tegum product for any two point sets, by placing them in hyperplanes intersecting in exactly one point 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.

Example
The pentagonal tegum is the tegum product of a pentagon and a line segment.

These are placed with matching centres, orthogonal to each other. Over this is stretched a sheet, that is held in position by the surface of the pentagon and line. The pentagon-surface is its vertices and edges. The line-surface is simply the two ends. These are the exposed parts when the pentagon and line are placed in a 2d and 1d space respectively.

The surface of the tegum consists of triangles, here formed by the pyramid product of the pentagon-edges and the line-ends. The body of the sheet does not contain any further parts of the pentagon or line, instead it consists of stretching between the surfaces of these figures. A line exists for each point of the surface of the pentagon (ie points on the edge) to points on the surface of the line (ie its ends).

Being a product of draught (drawing of something, like glass or chewing-gum), the effect of drawing the nullitope of one figure and something in the second figure, preserves the something in the second figure. So the surface (edges) of the pentagon and (ends) line are preserved, but the interior is lost.