Message #2751

From: Roice Nelson <roice3@gmail.com>
Subject: The exotic {4,4,4}
Date: Sat, 10 Aug 2013 16:10:23 -0500

Hi all,

Check out a new physical model of the exotic {4,4,4} H³ honeycomb!

http://shpws.me/oFpu

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}<http://games.groups.yahoo.com/group/4D_Cubing/message/1226>,
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:

http://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane#Regular_hyperbolic_tilings

The {∞} polygons are inscribed in
horocycles<http://en.wikipedia.org/wiki/Horocycle> (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!)