# Message #2754

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] The exotic {4,4,4}

Date: Sun, 11 Aug 2013 17:19:18 -0700

Here’s a slightly less awful sketch:

http://groups.yahoo.com/group/4D_Cubing/photos/album/1962624577/pic/40875152/view/

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}

> <groups.yahoo.com/group/4D_Cubing/photos/album/1962624577/pic/908182938/view/>

> to MagicTile something like this:

> groups.yahoo.com/group/4D_Cubing/photos/album/1962624577/pic/908182938/view/

>

> -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!

>>

>> 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!)

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