# Message #2758

From: schuma <mananself@gmail.com>

Subject: Re: The exotic {4,4,4}

Date: Tue, 13 Aug 2013 01:23:52 -0000

Neat. Just solved {32,3}3C with two moves… That’s a good appetizer for dinner.

Nan

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> 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@…>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@…>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} HÂ³ 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 {â,â} tiling and {p,q} tilings that approach it by

> >> increasing p or q. Check out the progression that starts with an {â,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 {â} 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 {â,â} tile loses its center. A horocycle has a

> >> single center on the boundary, so the inscribed {â,q} tiles have a clear

> >> center, but because an {â,â} 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 {â,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 {â,â}

> >> 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!)

> >>

> >>

> >>

> >>

> >>

> >>

> >

> >

> >

> >

> >

>