To: geometry-research@forum.swarthmore.edu
Date: 13 Dec 1995 09:50:37 -0500
From: Douglas Zare <zare@cco.caltech.edu>
Organization: Caltech
Subject: Polyhedra Which Tile Hyperbolic 3-Space
Which polyhedra are Vornoi cells of cosets of co-compact subgroups of
the isometries of H^3? What other polyhedra tile H^3? Is it still
open whether the Dehn invariant of any polyhedron tiling H^3 is 0?
Thanks,
Douglas Zare
http://www.cco.caltech.edu/~zare
Date: Wed, 13 Dec 1995 11:26:55 -0500 (EST)
From: John Conway <conway@math.princeton.edu>
Subject: Re: Polyhedra Which Tile Hyperbolic 3-Space
To: Douglas Zare <zare@cco.caltech.edu>
On 13 Dec 1995, Douglas Zare wrote:
> Which polyhedra are Vornoi cells of cosets of co-compact subgroups of
> the isometries of H^3?
There is absolutely no hope of giving any reasonable kind of
answer to this question; there is a plethora of possible groups,
and each group has a continuum of orbits, which can lead to a
variety of Voronoi polyhedra.
Even in the Euclidean case, the problem is not likely to be
solved very soon, although there it's pretty obvious that the
number of combinatorial types is bounded. There is an example by
Engels of a Voronoi cell that has (I think) 38 faces.
> What other polyhedra tile H^3?
This is probably even more hopeless. There are easy examples
that show that the ARE some more.
> Is it still
> open whether the Dehn invariant of any polyhedron tiling H^3 is 0?
I don't know, but expect so. This is not so hopeless - a
counterexample tiling must have certain slightly paradoxical
properties, and it's quite hopeful that one could use these
to disprove its existence, so proving that the invariant must be 0.
On the other hand, of course, if it needn't be, this could of course
be proved by exhibiting just one special tiling.
The paradoxical nature of the required tiling reminds me of
another problem - that of defining the density of a sphere-packing
in hyperbolic space. I have read of a "counterexample packing"
of spheres with the following properties:
i) There are two tilings T1 and T2 of the space into polyhedral
cells that each contain just one of the spheres.
ii) Each of T1 and T2 admits an automorphism group of isometries
that act transitively on its cells (so that all the cells of Ti
have the same volume, Vi).
iii) The volumes V1 and V2 are different (and both finite).
Does anyone know a reference for this?
John Conway
To: geometry-research@forum.swarthmore.edu
Date: 15 Dec 1995 14:02:05 -0500
From: Douglas Zare <zare@cco.caltech.edu>
Organization: The Geometry Forum
Subject: Re: Polyhedra Which Tile Hyperbolic 3-Space
John Conway (conway@math.Princeton.EDU) wrote:
>On 13 Dec 1995, Douglas Zare wrote:
>> Which polyhedra are Vornoi cells of cosets of co-compact subgroups of
>> the isometries of H^3?
>
> There is absolutely no hope of giving any reasonable kind of
>answer to this question; there is a plethora of possible groups,
>and each group has a continuum of orbits, which can lead to a
>variety of Voronoi polyhedra.
I asked a bit too generally, but what can be said to characterize
such polyhedra? I'm more interested in the case where the quotient
is a manifold. Can one recover the manifold or even the center from
a polyhedron? What can be said about the elements of the
fundamental group which are represented as faces?
> Even in the Euclidean case, the problem is not likely to be
>solved very soon, although there it's pretty obvious that the
>number of combinatorial types is bounded. There is an example by
>Engels of a Voronoi cell that has (I think) 38 faces.
>
>> What other polyhedra tile H^3?
>
> This is probably even more hopeless. There are easy examples
>that show that the ARE some more.
I suppose the higher-genus and really small ones wouldn't work. The
only way I know of getting arbitrarily small polyhedra which tile
is by tiling the gap between two horospheres. Are there essentially
different tilings?
>> Is it still
>> open whether the Dehn invariant of any polyhedron tiling H^3 is 0?
>
> I don't know, but expect so. This is not so hopeless - a
>counterexample tiling must have certain slightly paradoxical
>properties, and it's quite hopeful that one could use these
>to disprove its existence, so proving that the invariant must be 0.
Well, I just typed out what I was trying to make work, but I think
I just found a counterexample:
Consider the (x,y,z+) upper-half plane model and the convex hull
of the vertices (+-1,+-1,2), (+-1,+-1,1), (0,+-1,1), (+-1,0,1), and
(0,0,1). In other words, a 2x2 square in the height 2 horosphere
and 4 1x1 squares in the height 1 horosphere. Of course, the top
square is congruent to the bottom squares. By translations in 2Zx2Z,
this shape tiles a region around the horosphere of height 2. This
region is congruent to regions around the horospheres of height 2^n
for all integers n; these tile H^3.
Now, this shape probably doesn't have Dehn invariant 0. But even if
it does, a small polygonal bump added to the top allows the same
bump to be taken 4 times out of the bottom. So, choose the bump
to have nonzero Dehn invariant D, and the new shape will have Dehn
invariant -3D.
>On the other hand, of course, if it needn't be, this could of course
>be proved by exhibiting just one special tiling.
I hope the above works, but I've been up for some time so I actually
can't see it.
> The paradoxical nature of the required tiling reminds me of
>another problem - that of defining the density of a sphere-packing
>in hyperbolic space.
For foams in H^2 and H^3, I didn't find anything I could compute.
What happens if you take the lim sup of the density within balls
of increasing radius centered about each point p to get a function
of p, and then iterate using the average value instead of density?
>I have read of a "counterexample packing"
>of spheres with the following properties:
>
> i) There are two tilings T1 and T2 of the space into polyhedral
>cells that each contain just one of the spheres.
>
> ii) Each of T1 and T2 admits an automorphism group of isometries
>that act transitively on its cells (so that all the cells of Ti
>have the same volume, Vi).
>
> iii) The volumes V1 and V2 are different (and both finite).
That is disturbing if the spheres are in any way symmetric or
fairly large in comparison to V1 and V2. Otherwise, it still
disturbs me, but I suspect that frequently there exists a matching
between the sets of tiles of two tilings given that the tiles have
large diameter with respect to the ratio of their volumes, where a
tile of T1 is adjacent to a tile of T2 iff their intersection
contains a ball of radius r.
> Does anyone know a reference for this?
>
> John Conway
Thanks,
Douglas Zare
http://www.cco.caltech.edu/~zare