Blog:Polytopes in other spaces

Recall that a concrete polytope is an abstract polytope $$\mathcal P$$ together with a map $$f:\text{vert}(\mathcal P)\to X$$ from its rank 0 elements into some arbitrary set. So far, we've just been implicitly assuming that we're working with $$\mathbb R^n$$, or at most with $$\mathbb F^n$$ for $$\mathbb F\subseteq\mathbb R$$ (see conjugates). Well... sometimes we work on the sphere or hyperbolic space for the heck of it. But as it turns out, a lot of definitions carry over into much more general mathematical structures.

Let's start simple. To make sense of regular concrete polytopes, we only need that $$X$$ has some sort of structure we can preserve by. If $$X$$ is a like $$\mathbb R^n$$, this structure might be distance. In this case, we recover the usual definition. For an or for a, this structure might be collinearity. Perhaps it might be possible to sensibly embed polytopes on manifolds? Really, the possibilities are endless on this one.

Even though the usual definition for uniformity specifies all edge-lengths to be the same, you could forgo this by defining regular polygons to be uniform, and defining higher-dimensional uniforms as vertex-transitive polytopes with uniform facets. There's even more unexplored territory here than there is with regulars.

Unless you're working with Petrials or something similarly as weird, you probably want to have some notion of planarity. Perhaps the most general way to do this is to make $$X$$ a vector space. That still leaves an awful lot of choices for the base field though. Polytopes on a finite field sound fun. Also, what happens if you try to embed a concrete polytope in something like complex space? No, I don't mean complex polytopes. There might be some weird stuff to uncover there.

What if you want convexity? That seems to require nothing more than an ordered geometry. You can define the convex cover of a set as all points between two others in the set, and the convex hull as the union of the n-fold convex cover for any n. However, making polytopes out of these seems to once again require a vector space to define the face lattice. So overall, you'd need a vector space over an ordered field. There's a few of those, though they're all either subfields of the real numbers or some really weird stuff. Might be worth a try though.

I have the hunch that volume can be defined for planar polytopes in any vector space. However, I haven't proved this to be the case (I haven't even proved that the definition works at all!)

So overall, there seem to be a lot of directions we could generalize the study of concrete polytopes, mostly with vector spaces. Just imagine, enumerating uniform polyhedra in some vector space over a finite field, doesn't that sound fun? Hopefully I haven't cast a net that's hopelessly big here.