# Message #2760

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] The exotic {4,4,4}

Date: Tue, 13 Aug 2013 11:05:00 -0500

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@superliminal.com>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@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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