Notions and Notations.

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Re: Four Products, etc

Postby mr_e_man » Mon Jan 31, 2022 7:03 pm

There is another product (or something like a product), involving digons.

For two abstract polytopes A and B, let A'=A\{body(A)} and B'=B\{nulloid(B)}, and take the (disjoint) union C=A'∪B'. Define a partial order on C in terms of the partial orders on A and B, thus: a₁≤a₂ in C if and only if a₁≤a₂ in A'; b₁≤b₂ in C if and only if b₁≤b₂ in B'; and a≤b and not b≤a (where a is anything in A', and b is anything in B'). So, each vertex of B takes the place of the body of A.

The result C is always a polytope, if A and B are greater than (-1)-polytopes. The dimension is dim(C)=dim(A)+dim(B). Every [dim(A)-2, dim(A)+1]-subpolytope in C is a digon.

If B is a 1-polytope, then C is a ditope. If A is a 1-polytope, then C is a hosotope.
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Re: Notions and Notations.

Postby wendy » Wed Jan 25, 2023 10:41 am

Can you write this as an actual constuction of a polytope, and not in some mathematical puree.

It is listed as a product, but you describe it as a sum. A polytope product of A and B is the outer product of some subset of these (ie a set of all members of A, B, of the surface and some additional points), not a union.

The union of two sets (A, B, C) and (C, D, E) is the set (A, B, C, D, E). These five elements do not consist or comprise the outer product (AC, AD, AE, BC, BD, BE, CC, CD, CE). Further more, you have not indicated exactly how these set members would be realised in actual polytopes.
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