In its application to polytopes it is quite closely related to the hull operator.
Infact, we meanwhile know of 2 such applications, the tegum sum and the tegum product. While the tegum sum refers to the "cover" of 2 or more elements within the same space, the tegum product refers to such a "cover" of 2 or more elements within orthogonal spaces. More rigorously and formally, if P and Q would be any 2 polytopes those tegum terms could be defined as
- tegum sum(P, Q) = hull(P + Q) = hull( P ∪ Q ) - thus we need dim(P) = dim(Q) here.
- tegum product(P, Q) = hull(P ⊕ Q) = hull( (P,0) + (0,Q) )
The easiest example here for sure is the tegum product of N equivalent line segments. That one then is nothing but the N-dimensional crosspolytope. - Right in that example it becomes apparent, that the size of these line elements and the size of the lacing edges differs by a factor of sqrt(2). And so in general, the size of the lacing edges highly depends on the overall size of the respective factors or addends.
Student91 once asked for a new lace term, which he, in his first posts, termed "&#U" in the reading of "lace union". That one later settled in "&#zx", kind a lace prism with zero height. But, infact, in more formal rigourness, it is rather the following
- Let P and Q be the components of some compound, e.g. P = q3o3o and Q = o3o3q, thus comp(P, Q) = qo3oo3oq (a stella octangula);
- Provided that compound gives rise to a hull polytope hull(P, Q) = tegum sum(P, Q), where all lacing edges have some specific (say unit) size,
- Then we are allowed to write comp(P, Q)&#zx, e.g. qo3oo3oq&#zx (which one is nothing but the unit cube).
It should be noted that there is not a restriction here to all vertex pairings. Only those will be considered, which provide an outside visible lacing edge of the hull of the compound components, i.e. the true edges. Whereas those false lacing "edges", which would become internal, do not contribute to that restriction!
From the hull operation it becomes apparent as well, that way not all subelements of the compound components would contribute as subelements of the resulting polytope. Esp. the component polytopes themselves usually are "pseudo elements" only. (Only when one compound component is completely internal to the other, then the larger component "survives" as hull.)
For completeness it should be added, that we usually consider Dynkin describable polytopes, i.e. the Wythoffians. Those thus are all uniform and therefore circumscribable. Thus the compounding procedure here becomes much more definite: both the circumcenters will have to coincide and moreover the describing symmetry ought be the same for such a compounding description like in qo3oo3oq.
From a more abstract point of view, here we just have given a constructive definition for new sets of polytopes, the tegum product and the tegum sum, on the base of some formerly known polytopes P and Q. Esp. the restricting construction asking additionally for equal sized lacings only ("&#zx") is, from a mere mathematical point of view, rather "wild".
Therefore here a task, open to be solved:
- Could those "Student91-ian lace tegums" ("&#zx") could be classified outside from their construction device?
- Esp. could one start (and luckily also end) an exhaustive enlisting of those?
Beside the more theoretical aspect of this post (and its supposed answers) I further hope to get in here - in this new thread - a broard collection of already found such polytopes (resp. the polytopal descriptions thereof).