# Message #2752

From: schuma <mananself@gmail.com>

Subject: Re: The exotic {4,4,4}

Date: Sun, 11 Aug 2013 18:26:13 -0000

Summary of the long post:

Roice reinvented broccoli.

Remarkable!

A thought on the horocycle of {infinity, infinity} is that when the horocycle becomes the boundary of the disk itself, technically the horocycle does not exist any more. The boundary is not part of the Poincare disk model. So it’s natural not to talk about the "center" of a non-existing horocycle.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> Hi all,

>

> Check out a new physical model of the exotic {4,4,4} HÂ³ 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 {â,â} tiling and {p,q} tilings that approach it by

> increasing p or q. Check out the progression that starts with an {â,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 {â} 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 {â,â} tile loses its center. A horocycle has a

> single center on the boundary, so the inscribed {â,q} tiles have a clear

> center, but because an {â,â} 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 {â,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 {â,â}

> 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!)

>