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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