# Message #2756

From: Melinda Green <melinda@superliminal.com>

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

Date: Sun, 11 Aug 2013 20:32:30 -0700

Hello Roice,

I’m glad that you think that this puzzle makes sense. Also, I like your

idea of using fundamental domain triangles. As for other colorings (and

topologies), I would first hope to see the simplest one(s) first. This

3-coloring seems about as simple as possible though perhaps one could

remove an edge or two by torturing the topology a bit. As for

incorporating into MT versus creating a stand-alone puzzle, I have a

feeling that there might be some clever ways to incorporate it. One way

might be to implement it as a {N,3} for some large N. If a user were to

pan far enough to see the ragged edge, so be it. If it must be a

stand-alone puzzle, it might allow for your alternate colorings and

perhaps other interesting variants that would otherwise be too difficult.

-Melinda

On 8/11/2013 8:06 PM, Roice Nelson wrote:

>

>

> The puzzle in your pictures *needs* to be made!

>

> It feels like the current MagicTile engine will fall woefully short

> for this task, though maybe I am overestimating the difficulty. Off

> the cuff, an approach could be to try to allow building up puzzles

> using fundamental domain triangles rather than entire tiles, because

> it will be necessary to only show portions of these infinite-faceted

> tiles. (In the past, I’ve wondered if that enhancement is going to be

> necessary for uniform tilings.) It does seem like a big piece of

> work, and it might even be easier to write some special-case code for

> this puzzle rather than attempting to fit it into the engine.

>

> I bet there is an infinite set of coloring possibilities for this

> tiling too.

>

>

>

> On Sun, Aug 11, 2013 at 7:19 PM, Melinda Green

> <melinda@superliminal.com <mailto:melinda@superliminal.com>> wrote:

>

>

>

> 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}

>> <http://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/

>> <http://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!)

>>

>

>

>

>

>

>

>