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>