These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the **{6,3,7} Coxeter group**, by which I mean the one coming from the Coxeter diagram

$$\circ-6-\circ-3-\circ-7-\circ$$

Does this group act on a 4-dimensional real vector space equipped with a quadratic form of signature $++--$, with the four standard generators acting as reflections?

More generally: it's easy to learn about Coxeter groups acting as reflections on real vector spaces with a quadratic form of signature $(n,1)$: these 'hyperbolic' Coxeter groups show up as symmetries of hyperbolic honeycombs, and there's a lot of literature available on those. But what about 'ultrahyperbolic' Coxeter groups, which act as isometries of real vector spaces with quadratic form of signature $(n,m)$ with $n, m \ge 2$? Where can we learn about those?

Here is what Roice Nelson wrote about the above pictures, as part of a discussion about a puzzle I posed on Google+:

I just uploaded two pictures of the {6,3,7} to wikipedia:

https://commons.wikimedia.org/wiki/File:H3_637_UHS_plane_at_infinity_view_1.png

https://commons.wikimedia.org/wiki/File:H3_637_UHS_plane_at_infinity_view_2.png

The images look nothing like the {6,3,6}, so here's a little about what's going on and why I drew them differently...

When Coxeter extended the enumeration of honeycombs from 4 to 15, he did so by allowing the fundamental simplex to have "ideal" vertices. But he still limited the volume of the simplex to be finite. This restriction is equivalent to requiring the cells and vertex figures to be spherical and/or Euclidean. For a {p,q,r} honeycomb, $(p-2)(q-2) \le 4$ and $(q-2)(r-2) \le 4$.

But we can extend the enumeration of honeycombs to be endless if we allow the fundamental simplex to have infinite volume (to be noncompact). This means that one or more vertices of the simplex disappear completely, living beyond the plane at infinity. I've seen such points described in the literature as "ultra-ideal". Points outside the hyperbolic models do have geometric meaning btw (see Thurston's

Three Dimensional Geometry and Topologyp. 71). Noncompact honeycombs have cells and/or vertex figures that are hyperbolic tilings.In the progression above, note how the cells appear to fill more and more of hyperbolic space. For the first 3, we can see vertices inside H3, but for the {6,3,6}, the vertices have become ideal and each {6,3} cell is inscribed in the whole of hyperbolic space. Continuing to {6,3,7}, we might expect the vertices to continue their expansion, and indeed they do become ultra-ideal.

If we were to draw the {6,3,7} honeycomb in the same style, the edges would never meet, so we'd see a bunch of disconnected edges all over the place. Instead what the wiki images show is how the honeycomb interacts with the plane at infinity. It is a single slice of the honeycomb with that plane, in the upper half space model.

The vertex figure of a {6,3,7} honeycomb is a {3,7} tiling. What we see in the slice is a {3,7} tiling for every ultra-ideal vertex of the honeycomb. This arises because the cells haven't met at a vertex yet (even at infinity). Furthermore, we see these vertices laid out in {6,3} patterns. I've included two views, but it is the same honeycomb. The second may be easier to digest because the hexagonal pattern of vertices is more typical.

What would the {6,3,6} look like in a similar slice view? Well, those cells meet at ideal points on the boundary, and the points densely fill the plane at infinity, so it would just look solid black :)

The {6,3,7} puzzle is difficult in my eyes because honeycombs with {6,3} cells are already strange to get a handle on. Plus some formulas in Coxeter's paper are useless (e.g. the formula for in-radius already results in infinity for the {6,3,6}, and so doesn't help for {6,3,7}). Some easier noncompact honeycombs to think about first are the {3,3,7} and {7,3,3}.

Sorry, this comment is getting much longer than I wanted, and is probably opaque as dirt, so I think I'll stop for now. Henry Segerman and I have started preparing a math-art paper to share some of these images, and discuss this topic in more depth, hopefully more clearly!

I don't understand how all this relates to my conjecture that {6,3,7} acts on a 4d vector space with quadratic form of signature $++--$.

othercases. $\endgroup$ – John Baez May 6 '14 at 6:02Reflection Groups and Coxeter Groups.Incidentally, this representation is also always faithful. $\endgroup$ – Qiaochu Yuan May 6 '14 at 18:452more comments