# Message #2757

From: Roice Nelson <roice3@gmail.com>

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

Date: Mon, 12 Aug 2013 19:37:40 -0500

Hi Melinda,

I liked your idea to do large N puzzles, so I configured some biggish ones

and added them to the download :) They are in the tree at "Hyperbolic ->

Large Polygons". They take a bit longer to build and the textures get a

little pixelated, but things work reasonably well. Solving the {32,3} 3C

will effectively be the same experience as an {inf,3} 3C, though I would

still like to see the infinite puzzle someday too. One strange thing about

{inf,3} will be that no matter how much you hyperbolic pan, you won’t be

able to separate tiles from the disk boundary, whereas in these puzzles you

can drag a tile across the disk center and to the other side.

Download link:

http://www.gravitation3d.com/magictile/downloads/MagicTile_v2.zip

And some pictures:

http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/435475920/view

http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/810718037/view

http://groups.yahoo.com/group/4D_Cubing/photos/album/1694853720/pic/600497813/view

seeya,

Roice

On Sun, Aug 11, 2013 at 10:32 PM, Melinda Green <melinda@superliminal.com>wrote:

>

>

> 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>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/

>>

>> -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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