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!)
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