Crenellated quad hyperropes

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

Crenellated quad hyperropes

Postby polymath » Fri Jun 04, 2010 4:03 am

Hi! I'm new here. I've been investigating a family of polytopes I call crenellated quad hyperropes (CQHRd). They are a sequence of polytopes; one for each dimension d = 0, 1,2,... I am approaching this investigation from the shallow end, mathematically, based on my competence level. Still, I've been able to prove a number of interesting properties about this family. And, I'm still working on some of the proofs. I am nearing the point where I feel I can approach a mathematician to co-author an article on this.

Some of the interesting properties of CQHRd are:
1. It's possible (easy even) to draw a 2-d diagram partially (actually, substantially) representing each CQHRd.
2. CQHRd has the Fibonacci number of facets Fd+2
3. Analogous to the way that cross polytopes have all simplexes as proper elements, CQHRd have all crenellated quad hyperstrings (CQHS) as proper elements.
4. There are families of sets (of positive integers) having distinct subset sums that naturally relate to the family CQHRd.

The thing that is really amazing about these concerns the sequence of duals CQHR*d. Define the set of all elements of the sequence CQHR* as CQHR*e. CQHR*e is closed under rectangular product!
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Re: Crenellated quad hyperropes

Postby 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.
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Re: Crenellated quad hyperropes

Postby polymath » Fri Jun 04, 2010 5:44 pm

I'll throw out a brain teaser here that I will eventually come back to, as I've already worked out the answer to it. Quadtopes, as defined, feature quad faces (henceforth I will use the word quad instead of quadrilateral) which touch at at most one vertex. (I'm also using the Wikipedia polytope page naming convention, so 'face' is 2-d.) But, if there are two quad faces with a common vertex, they form a subpolytope of dimension 4. So, what kind of quadtope has a 3-d diagram consisting of the quad faces of a cuboctahedron?

In other words, there's a set of six quads in general position, each of which meets four other quads at its vertices in the same way the quad faces of a cuboctahedron do.

Clearly, the quadtope, if there is one, is not a cuboctahedron, because it must have a subpolytope of dimension 4. Can it exist? If so, what is the combinatorial type? Can you give its face lattice?
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Re: Crenellated quad hyperropes

Postby wendy » Sat Jun 05, 2010 7:37 am

These are nothing more than tegums.
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Re: Crenellated quad hyperropes

Postby polymath » Tue Jun 15, 2010 12:59 pm

Hello! I have just completed an 1,100 mile move.

Wendy K., regarding your statement that polygontopes are tegum products, I must disagree. The definitions are different: for a polygontope which is the free join convex hull of two polygons, the two polygons in general position can be disjoint (in which case they take up 5 dimensions), or they have a vertex in common (taking up 4 dimensions). For a tegum product of two polygons, the 2 polygons must be in general position beyond the fact that they have a point of intersection which is in the relative interior of each polygon. Hence, they must take up 4 dimensions. (So, the point of intersection isn't an element of either polygon, as it is for a polygontope.)
Last edited by polymath on Wed Jun 16, 2010 8:52 pm, edited 1 time in total.
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Re: Crenellated quad hyperropes

Postby wendy » Wed Jun 16, 2010 9:16 am

Conway discussed this join as well.

Still, the 'join' is as much a product as the cartesian product, and just as the cartesian product breaks into the comb and the prism products, so does the join break into the tegum and the prism product.

Pyramid products do indeed add a dimension: the simplex in N dimensions is the product of its vertices, meaning that the entire volume is added by the product.

Apart from the fact that the join product is coherent to the tegum-measure, i did not really see much point in it as a union.
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