# Message #2759

From: Melinda Green <melinda@superliminal.com>

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

Date: Mon, 12 Aug 2013 18:30:03 -0700

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 <mailto: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 <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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