by polymath » Fri Jun 04, 2010 2:08 pm
One motivation for studying CQHR's & CQHS's is that they belong to a class of of polytopes I call quadtopes, which in turn are a subset of polygontopes.
A polygontope is a polytope which is either a free join (i.e., convex hull) of a set of polygons in general position, with any two polygons intersecting in at most one vertex, or, a k-fold pyramid over such a free join convex hull of polygons. By general position, I mean that any two polygons in the set take up the maximum possible dimension. (I have seen this term used elsewhere in this sense.) So, for a polygontope, the convex hull (free join) of two disjoint polygons of the set is a five-dimensional polytope, and the free join convex hull of two polygons meeting at a vertex is a four-dimensional polytope. (The aforementioned polygons and the polytopes would, of necessity, be elements of the polygontope.)
Note: This definition describes a family of combinatorial types, and is intended to include the nullitope and simplexes via an empty set of polygons, and a k-fold pyramid over the nullitope.
Now, there is a certain arbitrariness to this definition, and I don't take lightly the responsibility of coining a new term. I chose this definition to define a class of polytopes that has features that make them easier to study than the terra incognita of polytopes in general, while still being interesting.
For example, why specify that any two polygons in the set have at most a vertex in common? Well, if the requirement were that the polygons must be disjoint, you would get a narrow and not terribly interesting class of polytopes. If you allowed two polygons to have (up to) an edge in common, you would include some polyhedra (including a 6-sided shape with six vertices that is almost a triangular prism, but has a diagonal edge bisecting one of the would-be quad faces). It would not, however, include a cube (even a crooked one), because opposite faces are not in general position. Someone might decide that such a class of polytopes is interesting; if so, they can call them 1-polygontopes. Then, the term polygontope is short for 0-polygontopes (allowing only intersections at 0-dimensional elements).
Why allow a k-fold pyramid over the free join convex hull? Well, a 2-fold pyramid over the free join convex hull is the same as the free join of the original set, plus a triangle with vertices at the two apices of the pyramid & one of the vertices of the (original) free join. It's awkward to allow 2-fold pyramids, but not 1-fold pyramids.
Quadtopes, of course, are defined by substituting quadrilateral for polygon in the definition above.
Last edited by
polymath on Wed Jun 16, 2010 9:02 pm, edited 1 time in total.