# Message #2292

From: schuma <mananself@gmail.com>

Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}

Date: Mon, 25 Jun 2012 06:05:03 -0000

Update:

{infinity, 3, 3} is added. You may want to check the shape of each cell,

{infinity,3}

<http://www.plunk.org/~hatch/HyperbolicTesselations/inf_3_trunc0_512x512<br>
.gif> , first (ignore the blue lines). I just noticed that this shape

looks like the biohazard symbol

<http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Biohazard_symb<br>
ol.svg/200px-Biohazard_symbol.svg.png> .

Because there is no loop here (the length of a loop is infinite), the

wireframe model of {infinity, 3, 3} is a tree.

Nan

— In 4D_Cubing@yahoogroups.com, Melinda Green <melinda@…> wrote:>>

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:> >> >

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} 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.> > -Melinda>