Message #2301

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] Hyperbolic Honeycomb {7,3,3}
Date: Tue, 03 Jul 2012 15:02:50 -0000

Natural parameter of ultrainfinite trangles is a distance between its sides (but radius of the inscribed circle also works). In H2 pictures with different parameters look equivalent, but faces of cells of different {3,3,n} honeycombs will have different parameters.

Andrey

— In 4D_Cubing@yahoogroups.com, Don Hatch <hatch@…> wrote:
>
> 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@…> 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@…
> > http://www.plunk.org/~hatch/
> >
> > ————————————
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
>
> –
> Don Hatch
> hatch@…
> http://www.plunk.org/~hatch/
>