Message #2291

From: Melinda Green <>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Sun, 24 Jun 2012 11:22:28 -0700

Wow. I understand very little of this but it is very cool stuff, Nan!
I’m guessing that the great geometer Coxeter didn’t spend much time on
this class of objects because life is finite and he had to draw the line
somewhere. It sounds like others have at least looked down this crazy
corridor but it is hard to know what has been studied or even published
on them. I understand that while he was alive, Coxeter was able to
answer these sorts of questions, at least being able to tell you who to
talk to, but now that he is gone, we’re a bit in the dark again,
unfortunately. It certainly sounds like there is a nice paper in here if
you are interested.

On 6/24/2012 1:14 AM, schuma wrote:
> Hi everyone,
> I’m continuing talking about my honeycomb/polytope viewer applet. I
> added a new honeycomb, and I think it deserves a new topic. This is
> {7,3,3}. Each cell is a hyperbolic tiling {7,3}. Please check it here:
> I first heard of this thing together with {3,3,7} in emails with Roice
> Nelson. He had been exchanging emails with Andrey Astrelin about them.
> We have NOT seen any publication talking about these honeycombs. Even
> when Coxeter enumerate the hyperbolic honeycombs, he stopped at
> honeycombs like {6,3,3}, where each cell is at most an Euclidean
> tessellation like {6,3}. He said, "we shall restrict consideration to
> cases where the fundamental region of the symmetry group has a finite
> content" (content = volume?), and hence didn’t consider {7,3,3}, where
> each cell is a hyperbolic tessellation {7,3}.
> We think {3,3,7} and {7 ,3,3} and other similar objects are
> constructable. I derived the edge length of {n,3,3} for general n, and
> then computed the coordinates of several vertices of {7,3,3}, then I
> plotted them. There’s really nothing so weird about this honeycomb. It
> looks just like, or, as weird as, {6,3,3}. The volume of the
> fundamental region of {7,3,3} may be infinite, but as long as we talk
> about the edge length, face area, everything is finite and looks normal.
> I could go and make {8,3,3}, {9,3,3} etc. I also believe {7,3,4} and
> {7,3,5} are also pretty well behaved, and looks just like {6,3,4} and
> {6,3,5} respectively. Or even {7,4,3}. As long as the vertex figure is
> finite (not like {3,4,4}), the image shouldn’t be crazy. Since we are
> facing an infinite number of honeycombs here, I feel I should stop at
> some point. After all we don’t understand {7,3,3} well, which is the
> smallest representative of them. I’d like to spend more energy making
> sense of {7,3,3} r ather than go further.
> 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?
> I still suspect people have discussed it somewhere in literature. But
> I haven’t found anything really related. Roice found the following
> statement and references. I don’t haven’t check them yet.
> __________
> I che cked ‘Abstract Regular Polytopes’, and was not able to find
> anything on the {7,3,3}. H3 honeycombs make several appearances
> at various places in the book, but the language seems to be
> similar to Coxeter, and their charts also limited to the same
> ones. On page 77, they distinguish between "compact" and
> "non-compact" hyperbolic types, and say:
> Coxeter groups of hyperbolic type exist only in ranks 3 to 10, and
> there are only finitely many such groups in ranks 4 to 10. Groups
> of compact hyperbolic type exist only in ranks 3, 4, and 5.
> But as best I can tell, "non-compact" still only refers to the
> same infinite honeycombs Coxeter enumerated. They reference the
> following book:
> J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge
> University Press (Cambridge, 1990).
> When researching just now on wikipedia, the page on uniform
> hyperbolic honeycombs ha s a short section on noncompact
> hyperbolic honeycombs, and also references the same book by
> Humphreys. So maybe this book could be a good reference to dig
> up, even though I suspect it will still not mention the {7,3,3}.
> Also: Abstract Regular Polytopes, p78:
> For the general theory of hyperbolic reflexion groups, the reader
> is referred to Vinberg [431-433]. We remark that there are
> examples of discrete groups generated by hyperplane reflexions in
> a hyperbolic space which are Coxeter groups, but do not have a
> simplex as a fundamental region.
> These honeycombs fall into that category.
> Here are those references:
> [431] E. B. Vinberg, Discrete groups in Lobachevskii spaces
> generated by reflections, Mat. Sb. 72 (1967), 471-488 (= Math.
> USSR-Sb. 1 (1967), 429-444).
> [432] E. B. Vinberg, Discrete linear groups generated by reflec
> tions, Izv. Akad. Nauk. SSSR Ser. Mat. 35 (1971) 1072-1112 (=
> Math. USSR-Izv. 5 (1971), 1083-1119).
> [433] E. B. Vinberg, Hyperbolic reflection groups. Uspekhi Mat.
> Nauk 40 (1985), 29-66 (= Russian Math. Surveys 40 (1985), 31-75).
> ______________
> Now I can only say "to the best of our knowledge, I haven’t seen any
> discussion about it".
> Some more thoughts by Roice:
> __________
> We know that for {n,3,3), as n -> 6 from higher values of n, the
> {n,3} tiling approaches a horosphere, reaching it at n = 6.
> For {7,3,n), as n -> infinity, does the {7,3} tiling approach a
> horosphere as well? The curvature definitely flattens out as n
> increases. If cells are a horosphere in the limit, a
> {7,3,infinity} tiling would have finite cells. It would have an
> infinite edge-figure, in addition to an infinite vertex-figure,
> but as Coxeter d id an enumeration allowing the latter, why not
> allow the former? I’d like to understand where in Coxeter’s
> analysis a {7,3,infinity} tiling does not fit in. One guess is
> that even if the {7,3} approaches a horosphere, it’s volume also
> goes to 0, so is trivial. The heptagons get smaller for larger n,
> so I suppose they must approach 0 size as well.
> It would also be interesting to consider how curvature changes for
> {n,3,3} as n-> infinity, especially since we already know what the
> {infinity,3} tiling looks like.
> _______________
> Currently I can’t imagine what {7,3,n} like when n>=6. So I really
> cannot comment on {7,3,infinity}. But {infinity, 3, 3} seems to be a
> good thing to study.
> My formula for the edge length of {n,3,3} is as follows. Following
> Coxeter’s notation, if 2*phi is the length of an edge of {n,3,3}
> (n>=6), then
> cosh(2*phi) = 3*cos^2(pi/n) - 1
> Sanity check: when n = 6, this formula gives cosh(2*phi)=5/4, which is
> consistent with the number in Coxeter’s table: cosh^2(phi)=9/8.
> By sending n to infinity, the edge length of {infinity, 3, 3} is
> arccosh(2). I should be able to plot it soon.
> By the way, in the applet there’s a "Clifford Torus". It looks much
> more beautiful than the polytopes, because the colors of the edges
> work pretty well here. Imagine you can fly around a donut, or go into
> the donut. The amazing thing is if the space is 3-sphere, the view
> inside the donut is exactly as same as the outside.

Heh, yes, the moment you think you are punching through to the interior
of the donut you suddenly find yourself outside again! This is great
stuff, Nan. I encourage you to keep going as far as your interest takes you.