Message #2298

From: Don Hatch <hatch@plunk.org>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Tue, 03 Jul 2012 01:25:02 -0400

Hi Roice,

Yeah, I was wondering if there’s a meaningful interpretation
of the complex edge lengths too.
I was thinking maybe it’s more helpful to just look
at cosh(half edge length) instead of the edge length itself…
that would be a pure imaginary number instead of a complex number,
which is a *little* easier to think about…
and then I was thinking maybe sinh(half edge length) might be
more meaningful than cosh(half edge length)…
I think that’s something akin to a half-chord length,
analagous to sin(half edge length) of a spherical tiling
(though I still have a lot of trouble visualizing what that means
in the hyperbolic case).

Sure, I’m interested in what you guys came up with
along the lines of a {3,ultrainfinity}…
I guess it would look like the picture Nan included in his previous e-mail
(obtained by erasing some edges of the {6,4})
however you’re free to choose any triangle in-radius
in the range (in-radius of {3,infinity}, infinity], right?
Is there a nicer parametrization of that one degree of freedom?
Or is there some special value which could be regarded as the canonical one?

Don

On Sat, Jun 30, 2012 at 12:39:16PM -0500, Roice Nelson wrote:
>
>
> Hi Don,
> Thanks for your enlightening email, and for correcting some speculations I
> made without thinking deeply enough.
> * I was totally wrong about horosphere cells being finite.
> * When considering {7,3,n} as n increases, I see it was incorrect to
> conclude the heptagon size decreases (some flawed internal reasoning).
> Interesting that the magnitude of the complex edge length starts
> decreasing when n>=7, although I guess those complex outputs are
> pretty meaningless (then again, maybe not!). Since I got the trend
> backwards for low n, I had no idea about the vertices going to
> infinity at n = 6. I should have noticed that {3,3,6}, {4,3,6},
> {5,3,6}, and {6,3,6} all do the same thing. It’s noteworthy that the
> "vertices -> infinity" pattern holds for any {p,3,6}, which makes
> sense since the vertex figure is an infinite tiling.
> 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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–
Don Hatch
hatch@plunk.org
http://www.plunk.org/~hatch/