Wednesday, December 24, 2014
Naively, a polytope is a collection of regions. This concept of polytope as a collection of non-overlapping convex regions has the advantage of specifying the surface. However, I would prefer to define a polytope as a collection of qualities, such as facet relations, convexity, and colinearity. Rather than list and define every quality of polytopes, it is easier to define transformations of region sets, and define an invariant for those transformations. Given a set of regions, I can add a boundary and the shape of the result is unchanged. Similarly, I can move a boundary, and even cause a migration of the underlying space without changing the shape of the polytope embedded in the space. Start with a partition of the regions in the polytope that excludes unnecessary boundaries, and specifies which portions of the boundaries are relevant. This sort of partition is explained in the prior post, "Polytopes, Convexity, Colinearity". Consider the power set of the partition. Choose those elements of the power set that form (convex) super-regions. Each of these elements has well defined sidedness wrt a subset of boundaries in the original space. Define a tristate space as a sidedness space where some sidednesses are left undefined. Map the chosen power set elements to regions in a tristate space. I claim this tristate space is invariant as required. It remains to pick a few polytope qualities and prove that region sets different in those qualities have different tristate spaces. Deferring that task, I'd prefer to identify conditions upon tristate spaces sufficient to embed them into linear spaces. Then I could count all possible polytopes of a given dimension and complexity. Define a tristate subspace as a tristate space with a set of boundaries removed, and identical regions combined. Define a linear subspace as a tristate subspace without undefined sidedness. Define a trispace subregion as one with a superset of defined sidedness. Read top-level region as one without subregions. I claim a tristate space is embeddable iff the sum of f(g(r),n) over all top-level regions r in any tristate subspace is equal to f(m,n), where m is the number of boundaries in the tristate subspace, f is the number of regions in a linear space, and g(r) is the number of boundaries in the tristate subspace r is undefined wrt. To prove this, construct the embedding by combining all linear subspaces of the tristate space. Because simplex spaces are unique up to automorphism, each boundary of the tristate space is included in a linear subspace. The resulting recombined space is linear because its subspaces are linear, and it is an embedding because its boundaries have the same sidedness.