Message #2294

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Sat, 30 Jun 2012 12:39:16 -0500

Hi Don,

Thanks for your enlightening email, and for correcting some speculations I
made without thinking deeply enough.

Offline, Nan and I also discussed the 2D analogue of the {3,3,7}, something
akin to the {3,inf} tiling where the triangle vertices are no longer
accessible. Let me know if you’re interested in that discussion, as it
would be cool to hear your thoughts.

Anyway, thanks again. Very cool stuff,
Roice


On Fri, Jun 29, 2012 at 7:32 PM, Don Hatch <hatch@plunk.org> wrote:

> 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:
> > 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?),
>
> 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:
>
> acosh(cos(pi/p)*sin(pi/r)/sqrt(1-cos(pi/q)^2-cos(pi/r)^2))
>
> {7,3,2} -> acosh(1.0403492368298681) = 0.28312815336765745
> {7,3,3} -> acosh(1.1034570002469741) = 0.45104488629937328
> {7,3,4} -> acosh(1.2741623922635352) = 0.72453736133879376
> {7,3,5} -> acosh(1.7137446255953275) = 1.1331675164780453
> {7,3,6} -> acosh(+infinity) = +infinity
> {7,3,7} -> acosh(+-1.5731951893240572 i) = (1.2346906773191777 +-
> 1.5707963267948966 i)
> {7,3,8} -> acosh(+-1.0714385881055031 i) = (0.9309971259601171 +-
> 1.5707963267948966 i)
> {7,3,9} -> 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
>
> –
> Don Hatch
> hatch@plunk.org
> http://www.plunk.org/~hatch/
>
>
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