Message #2290

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Sun, 24 Jun 2012 13:09:59 -0500

>
>
> It’s not clear for me whether we can identify some heptagons in {7,3} to
> make it Klein Quartic, in {7,3,3}. For example, in the hypercube {4,3,3},
> we can replace each cubic cell by hemi-cube by identification. The result
> is that all the vertices end up identified as only one vertex. I don’t know
> what’ll happen if I replace {7,3} by Klein Quartic ({7,3}_8). It will be
> awesome if we can fit three KQ around each edge to make a polytope based on
> {7,3,3}. If "three" doesn’t work, maybe the one based on {7,3,4} or {7,3,5}
> works. I actually also don’t know what’ll happen if I replace the
> dodecahedral cells of 120-cell by hemi-dodecahedra. Does anyone know?
>
>
There is the hemi-120-cell, made by identifying antipodal dodecahedra. It
has 60 dodecahedral cells (not hemi), and is a "projective regular
polytope". As far as a "locally projective" polytope based on the 120-cell
and hemi-dodecahedra, I have not seen such a thing, but "Abstract Regular
Polytopes" again has some information. The following quote is from the end
of section 14A, p509. For this quote, keep in mind the Petersen graph is
the graph of the hemi-dodec.

Finally, it is of course desirable to enumerate all the locally projective
> regular polytopes. As we have hinted at the beginning of this section, the
> famous Petersen graph occurs naturally when the Schlafli symbol consists of
> 3’s and 5’s. Peterson graphs have been extensively studied in graph theory
> and diagram geometries, and this work may be relevant in the present
> context. There are many examples of diagram geometries related to finite
> simple groups which are "locally Peterson" (see …). For example, from
> the locally projective polytope {{3,5}_5, {5,3}_5} with group PSL(2, 11),
> we obtain a locally Petersen diagram geometry of rank 3 by omitting the
> facets from the face poset. *We do not know whether there are other
> interesting regular polytopes which correspond to locally Petersen diagram
> geometries.* Similarly, the edge-graph of {{3,5}_5, {5,3}_5} is a graph
> which is locally Petersen (that is, the induced subgraph on the neighbours
> of each vertex is a Petersen graph). *The finite graphs which are
> locally Peterson have been completely described* (see [53, p. 37]).

(emphasis mine). The reference mentioned at the end is:

[53] A. E. Brouwer, A. M. Cohen, and A. Neumaier, *Distance-Regular Graphs*.
Springer-Verlag (New York-Heidelberg-Berlin, 1989).

As for polytopes that are locally the {7,3}_8, I’m in the dark as well. I
have no idea if such a thing is possible.

Roice