Message #2293

From: Don Hatch <>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Fri, 29 Jun 2012 20:32:14 -0400

Hi Nan,

I just wanted to address a couple of points that caught my eye
in your message (and in the part of Roice’s that you quoted)…

On Sun, Jun 24, 2012 at 08:14:31AM -0000, 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?),

Right. The fundamental region is the characteristic simplex,
so (since even ideal simplices have finite volume)
this is the same as saying that all the vertices
of the characteristic simplex (i.e. the honeycomb vertex, edge center,
face center, cell center) are "accessible" (either finite, or infinite
i.e. at some definite location on the boundary of the poincare ball).
So you’re examining some cases where that condition is partially
relaxed, i.e. the fundamental region contains more of the horizon than just
isolated points there… and the characteristic tetrahedron
is actually missing one or more of its vertices.

> 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.

{7,3,3} yes, in the sense that the vertices/edges/faces are finite,
and there’s clear local structure around them, and, as you observe,
the edge length formula works out fine
(but not the cell in-radius nor circum-radius formula)
and you can render it (as you have– nice!)…

{3,3,7} less so… its vertices are not simply at infinity (as in {3,3,6}),
they are "beyond infinity"…
If you try to draw this one, none of the edges will meet at all (not even at
infinity)… they all diverge! You’ll see each edge
emerging from somewhere on the horizon (although there’s no vertex
there) and leaving somewhere else on the horizon…
so nothing meets up, which kind of makes the picture less satisfying.
If you run the formula for edge length or cell circumradius, you’ll get, not infinity,
but an imaginary or complex number (although the cell in-radius is finite, of
course, being equal to the half-edge-length of the dual {7,3,3}).

It may be Coxeter refrained omitted these figures
because the "beyond infinity" parts are awkward to talk about,
and if you insist on running the formulas and completing the tables,
a lot of it will consist of imaginary and complex numbers
that aren’t all that meaningful physically, and might scare some readers away
(even though, as you’ve noted, some of the entries
are perfectly fine finite numbers or plain old infinity).

> 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} rather 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 checked ‘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 has 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 reflections,
> 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".

I noticed one further possible reference in Coxeter’s "reguar honeycombs in
hyperbolic space" paper– on the first page, he refers to:
"… (Coxeter 1933), not insisting on finite fundamental regions,
was somewhat lacking in rigour"
where (Coxeter 1933) is "The densities of the regular polytopes, Part 3".
I believe that paper is in the collection "Kaleidoscopes: Selected
writings of H.S.M. Coxeter" (my copy of which is buried in a box in
storage somewhere :-( ). I suspect that one *will* mention the {7,3,3};
I’d be interested to know what he says about it, now that we’re thinking
along those lines.

> 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.

Right… or more precisely,
the circumsphere, edge-tangency-sphere, face-tangency-sphere, and in-sphere
all approach horospheres
(different horospheres, but sharing the same center-at-infinity)…

> For {7,3,n), as n -> infinity, does the {7,3} tiling approach a
> horosphere as well?

I’m not completely confident that this will stay meaningful
as we lose the locations of the vertices (for n >= 7).
The circum-sphere certainly becomes ill-defined…
however one or more of the other tangency spheres
might stay well-defined.
One concievable outcome might be
that the in-sphere and mid-spheres approach different limits–
maybe the in-sphere approaches a horosphere but the face-tangency
mid-sphere doesn’t, and maybe the edge-tangency mid-sphere
is ill-defined just like the circumsphere is.

In thinking about this,
I have to first think about the significant
events that happen for smaller n…
{7,3,2} two cells, the wall between them tiled with {7,3}, cell centers are imaginary/complex
{7,3,3} finite vertex figure and local structure, although cell centers are imaginary/complex
{7,3,4} same
{7,3,5} same
{7,3,6} infinite vertex figure, vertices are at infinity (and cell centers still imaginary/complex)
{7,3,7} self-dual; both vertices and cell centers are now imaginary/complex
And now, like Nan, I’ve lost my intuition…
{7,3,7} is the one for me to ponder at this point.

> The curvature definitely flattens out as n
> increases.

right (i.e. the curvature increases, i.e. becomes less negative)

> If cells are a horosphere in the limit, a {7,3,infinity}
> tiling would have finite cells.

hmm? why?

> It would have an infinite edge-figure,
> in addition to an infinite vertex-figure, but as Coxeter did an
> enumeration allowing the latter, why not allow the former?

well, for things like {3,infinity} and {3,3,6}
with infinite vertex-figure, you can still draw them and measure things
about them even though the vertex figures are infinite– the vertices
are isolated, at least. it seems to me that if the edge figure is
infinite, then it can no longer have isolated vertices (if the vertices
are even accessible at all),
so it’s hard to draw a definite picture of anything any more,
or make any measurements… so we have less and less we can say about
the thing, I guess.

> 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,

Are you sure?
The edge length is finite for {7,3,2…5},
and infinite for {7,3,6}… that makes me think the heptagons are probably *growing*,
not shrinking, at least for n in that range…
and after that, the edge length is the acosh of an imaginary number,
so it’s hard to say whether it’s growing or shrinking or what.
To verify, the formula for the half-edge-length is:


&#123;7,3,2&#125; -&gt; acosh(1.0403492368298681) = 0.28312815336765745<br>
&#123;7,3,3&#125; -&gt; acosh(1.1034570002469741) = 0.45104488629937328<br>
&#123;7,3,4&#125; -&gt; acosh(1.2741623922635352) = 0.72453736133879376<br>
&#123;7,3,5&#125; -&gt; acosh(1.7137446255953275) = 1.1331675164780453<br>
&#123;7,3,6&#125; -&gt; acosh(+infinity) = +infinity<br>
&#123;7,3,7&#125; -&gt; acosh(+-1.5731951893240572 i) = (1.2346906773191777 +- 1.5707963267948966 i)<br>
&#123;7,3,8&#125; -&gt; acosh(+-1.0714385881055031 i) = (0.9309971259601171 +- 1.5707963267948966 i)<br>
&#123;7,3,9&#125; -&gt; acosh(+-0.8448884457716658 i) = (0.7673378247178905 +- 1.5707963267948966 i)

All that said, I still don’t have a clear picture of what happens
when n goes to infinity. We certainly lose the vertices
at n=7, so the circum-sphere isn’t well-defined…
and I think we must lose the edges eventually as well? in which case the
edge-tangency mid-sphere isn’t well-defined either…
but I’m guessing we *don’t* lose the face centers…
so the face-tangency sphere and in-sphere may still be well-defined,
and may approach a limit,
in which case if your question has meaning,
one of those limits would be its meaning (I think). And I don’t know the answer.

> 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.
> Nan


Don Hatch