# Message #2289

From: schuma <mananself@gmail.com>

Subject: Hyperbolic Honeycomb {7,3,3}

Date: Sun, 24 Jun 2012 08:14:31 -0000

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:

http://people.bu.edu/nanma/InsideH3/H3.html

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} 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".

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 did 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.

Nan