What are the equivalents of regular polytopes in finite or p-adic fields?

Question

I was reading about regular polytopes yesterday and it struck me that I don't know if they have equivalents in spaces that aren't powers of R. Presumably over each field, the regular polygons, polyhedra, etc can be separately classified and might be distinct. Are there any references on this?

Answer 1

I think the problem with defining regular polytope over other fields lies less in the notion of "regularity", but more in the notion of "polytope". At least how I am thinking about these, polytopes are convex, and a notion of convexity only exists if the field is ordered.

One alternative approach is to ignore the whole "convex set" thing, and consider a polytope purely as the set of its vertices. One has to be careful though, as already in $\Bbb R^n$ not all finite sets of points are vertex sets of (convex) polytopes (they have to be in "convex position").

This problem vanishes if we restrict to sufficiently symmetric polytopes, say, vertex-transitive polytopes. The vertices of a vertex-transitive polytope are the orbit of a finite matrix group, and conversely, any such orbit is the set of vertices of a veretx-transitive polytope (the convex hull of these points). And this now directly generalized to other fields.

Given a field $\Bbb F$, a finite matrix group $\Gamma\subset\mathrm{GL}(\Bbb F^n)$ and a point $v\in\Bbb F^n$, we can define the orbit polytope $\mathrm{Orb}(\Gamma,v)$ simply as the orbit of $v$ w.r.t. $\Gamma$ (in the case $\Bbb F=\Bbb R$ one would haev taken the convex hull of this orbit to obtain an actual convex polytope).

If you have an inner product, you might restrict to orthogonal matrix groups. Otherwise, you should consider two polytopes equivalent if they are related by an invertible linear transformation (rather than a rigid motion as in $\Bbb R^n$).

Since regular polytopes are vertex-transitive, this might be a valid approach. Now remains the part of generalizing "regularity". We do not have a notion of "face" for these orbit polytopes, so saying that all faces are congruent etc. will not work. However, regular polytopes over $\Bbb R$ are in a correspondence with the "finite reflection groups", and one can define these over general fields as well. Maybe this would be an approach, but I am not too familiar with this generalization.

Answer 2

The shortchords of the various polygons {x} have a value in the p-adic systems, if x divides p+1 or p-1. There is a separate p-adic number for each of the polygonal stars (ie there are euler(x)/2 solutions.

Because all of the other chords of the polygon are derived from C(n+1)= aC(n)-C(n-1), with C(0)=0, C(1)=1, it follows that under the conditions above, all of the chords of a polygon can be realised as a p-adic value.