Message #2762
From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] The exotic {4,4,4}
Date: Thu, 22 Aug 2013 14:52:07 -0000
Hi all,
I’ve tried 4 and 6 colors. As I expected, they are just versions of our familiar tetrahedron and cube. Unfortunately, I almost forgot how to solve them, so results are not very good: 60 and 142 twists…
I’ll think about coloring of {infinity,3} - it should be not difficult.
Andrey
— In 4D_Cubing@yahoogroups.com, Melinda Green <melinda@…> wrote:
>
> You mean this message?:
> http://games.groups.yahoo.com/group/4D_Cubing/message/1492 Perhaps I
> shouldn’t be surprised but I am. I’ve spent a little time with the other
> infinite puzzles without much progress so far. Has anybody else tried
> them? If so, do you feel that there is any difference between these and
> the rest of MT’s bestiary?
>
> Best,
> -Melinda
>
> On 8/13/2013 9:05 AM, Roice Nelson wrote:
> >
> >
> > I want to recall that Andrey pointed out all the 3C puzzles are
> > topologically the same, and behave like the {4,3} 3C (hemicube). The
> > state space does feel too small to really be enjoyable…even as an
> > appetizer :)
> >
> > Thanks so much for the suggestions Melinda. I especially like the
> > idea to cache the puzzle build data. Don’t know why I never thought
> > of that before! I added your thoughts to my running trello list.
> >
> > Cheers,
> > Roice
> >
> >
> >
> > On Mon, Aug 12, 2013 at 8:30 PM, Melinda Green
> > <melinda@… <mailto:melinda@…>> wrote:
> >
> >
> >
> > Very nice, Roice!
> >
> > I was pretty sure the {inf,3} wouldn’t be very difficult but I
> > didn’t expect it to be this easy It seems like god’s number for it
> > can be counted on one hand! One nit: Scrambling it with 1000
> > twists rings the "solved" bell a whole bunch of times as it
> > accidentally solves it self many times. Silencing the solved sound
> > during scrambling will be helpful, but then you should probably
> > also discard all the twists that led to it since it just becomes
> > unneeded log file baggage.
> >
> > The experience of scrolling around in the {32,3} is better than I
> > imagined. Somehow I expected to see only the fundamental polygons.
> > With N >= 100 you can probably just mask off the outermost few
> > pixels of the limit making it indistinguishable from the inf
> > version. You might also not center a face in the disk to disguise
> > its finiteness. Users can still scroll a face to the center but
> > they’d almost need to be trying to do that, and the larger the N,
> > the harder that will be.
> >
> > The ragged borders are indeed unsightly. Normally that’s not a
> > problem for solvers but it does go against the wonderful amount of
> > polish you’ve applied to MT in general. At the very least it shows
> > us what the experience can be which will be important in finding
> > out how interesting these puzzles are compared with other
> > potential puzzles.
> >
> > As for the time needed to initialize these puzzles, perhaps you
> > can cache all the build data for all puzzles so that you never pay
> > more than once for each? It might also be nice to ship with the
> > build data for whichever puzzle you make the default. One last
> > minor suggestion: If it’s not tricky, would you please see if you
> > can make the expanding circles animation spawn new circles
> > centered on the mouse pointer when it’s in the frame? That would
> > provide a nice distraction while waiting.
> >
> > Really nice work, Roice. Thanks a lot!
> > -Melinda
> >
> >
> > On 8/12/2013 5:37 PM, Roice Nelson 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@… <mailto: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@… <mailto: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/
> >>>> <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 {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!)
> >>>>
> >>>
> >>>
> >>>
> >>>
> >>
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