Message #2753

From: Melinda Green <>
Subject: Re: [MC4D] The exotic {4,4,4}
Date: Sun, 11 Aug 2013 16:34:04 -0700

Lovely, Roice!

This makes me wonder whether it might be possible to add a 3-color
to MagicTile something like this:


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!)