# Message #2770

From: Andrey <andreyastrelin@yahoo.com>

Subject: {10,3}, 6 colors? Re: [MC4D] The exotic {4,4,4}

Date: Mon, 26 Aug 2013 08:15:05 -0000

Hello, Roice,

I’ve tried to add new puzzle to Magic Tiles, but without success. The puzzle I’ve selected is non-oriented {10,3}, 6 colors (another version of hemidodecahedron), colors should be like this: http://groups.yahoo.com/group/4D_Cubing/photos/album/772706687/pic/2032129608/view . But all that I could get is a carpet with white spots (but they all are in proper places). What should I write in xml and why?

Andrey

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> 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@…>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@…>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!)

> >>>

> >>>

> >>>

> >>>

> >>>

> >>>

> >>

> >>

> >>

> >>

> >

> >

> >

> >

> >

>