Message #2754

From: Melinda Green <>
Subject: Re: [MC4D] The exotic {4,4,4}
Date: Sun, 11 Aug 2013 17:19:18 -0700

Here’s a slightly less awful sketch:

On 8/11/2013 4:34 PM, Melinda Green wrote:
> Lovely, Roice!
> This makes me wonder whether it might be possible to add a 3-color
> {inf,3}
> <>
> to MagicTile something like this:
> -Melinda
> On 8/10/2013 2:10 PM, Roice Nelson wrote:
>> Hi all,
>> Check out a new physical model of the exotic {4,4,4} H³ honeycomb!
>> Each cell is a tiling of squares with an infinite number of facets.
>> All vertices are ideal (meaning they live at infinity, on the
>> Poincare ball boundary). Four cells meet at every edge, and an
>> infinite number of cells meet at every vertex (the vertex figure is a
>> tiling of squares too). This honeycomb is self-dual.
>> I printed only half of the Poincare ball in this model, which has
>> multiple advantages: you can see inside better, and it saves on
>> printing costs. The view is face-centered, meaning the projection
>> places the center of one (ideal) 2D polygon at the center of the
>> ball. An edge-centered view is also possible. Vertex-centered views
>> are impossible since every vertex is ideal. A view centered on the
>> interior of a cell is possible, but (I think, given my current
>> understanding) a cell-centered view is also impossible.
>> I rendered one tile and all the tiles around it, so only one level of
>> recursion. I also experimented with deeper recursion, but felt the
>> resulting density inhibited understanding. Probably best would be to
>> have two models at different recursion depths side by side to study
>> together. I had to artificially increase edge widths near the
>> boundary to make things printable.
>> These things are totally cool to handle in person, so consider
>> ordering one or two of the honeycomb models :) As I’ve heard Henry
>> Segerman comment, the "bandwidth" of information is really high. You
>> definitely notice things you wouldn’t if only viewing them on the
>> computer screen. The {3,6,3} and {6,3,6} are very similar to the
>> {4,4,4}, just based on different Euclidean tilings, so models of
>> those are surely coming as well.
>> So… whose going to make a puzzle based on this exotic honeycomb? :D
>> Cheers,
>> Roice
>> As a postscript, here are a few thoughts I had about the {4,4,4}
>> while working on the model…
>> In a previous thread on the {4,4,4}
>> <>,
>> Nan made an insightful comment. He said:
>> I believe the first step to understand {4,4,4} is to understand
>> {infinity,
>> infinity} in the hyperbolic plane.
>> I can see now they are indeed quite analogous. Wikipedia has some
>> great pictures of the {∞,∞} tiling and {p,q} tilings that approach it
>> by increasing p or q. Check out the progression that starts with an
>> {∞,3} tiling and increases q, which is the bottom row of the table here:
>> The {∞} polygons are inscribed in horocycles
>> <> (a circle of infinite radius
>> with a unique center point on the disk boundary).
>> The horocycles increase in size with this progression until, in the
>> limit, the inscribing circle is*the boundary of the disk itself.*
>> Something strange about that is an {∞,∞} tile loses its center.
>> A horocycle has a single center on the boundary, so the inscribed
>> {∞,q} tiles have a clear center, but because an {∞,∞} tile is
>> inscribed in the entire boundary, there is no longer a unique center.
>> Tile centers are at infinity for the whole progression, so you’d
>> think they would also live at infinity in the limit. At the same
>> time, all vertices have also become ideal in the limit, and these are
>> the only points of a tile living at infinity. So every vertex seems
>> equally valid as a tile center. Weird.
>> This is good warm-up to jumping up a dimension. The {4,4,3} is kind
>> of like an {∞,q} with finite q. It’s cells are inscribed in
>> horospheres, and have finite vertices and a unique center. The
>> {4,4,4} is like the {∞,∞} because cells are inscribed in the boundary
>> of hyperbolic space. They don’t really have a unique center, and
>> every vertex is ideal. Again, each vertex sort of acts like a center
>> point.
>> (Perhaps there is a better way to think about this… Maybe when all
>> the vertices go to infinity, the cell center should be considered to
>> have snapped back to being finite? Maybe the center is at some
>> average of all the ideal vertices or at a center of mass? That makes
>> sense for an ideal tetrahedron, but can it for a cell that is an
>> ideal {4,4} tiling? I don’t know!)