# 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/

>

> ————————————

>

> Yahoo! Groups Links

>

>

>

>

–

Don Hatch

hatch@plunk.org

http://www.plunk.org/~hatch/