# Message #1244

From: Andrey <andreyastrelin@yahoo.com>

Subject: Re: {4,4,4}

Date: Wed, 03 Nov 2010 07:23:38 -0000

At first I thought about {3,infinity} as a puzzle with material overlapping twists. So each cell has 7 stickers (one 1C, three 2C and three infinite-C). Once you get periodic coloring of the puzzle (and there is a lot of them), vertex pieces will look as a periodic sequence of colors. For example, for 8-color scheme you’ll get 6 vertex pieces with colors (0132), (0264), (0451), (4675), (1573) and (2376). Actually, it will be {3,4} spherical puzzle. Also {3,infinity} will include {3,6}, {3,7} and all other similar objects :) BTW, {infinity,3} gives easy way to find coloring schemes for {N,3}. For example, there is 36-colors for {9,3}, but I’m not sure about 12- or 18-colors.

Alterantive way is to use deep-cut of {3,infinity} with movable centers of adjacent faces but unmovable (and non-rotating) vertex pieces. Or enable rotation of oricycles containing vertecies as well, but it will be different puzzle… What is {3,infinity} rectified? Something strange (like (3,infinity,3,infinity) polyhedron).

For {4,4,4} (that can be built from the checkerboard coloring of {4,4,3} by expanding of only black cells) I also think about twists with material overlapping. Vertex pieces there will have periodic 2D ({4,4}) coloring, and it will be difficult to describe their "position and orientation".

Slicing of {4,4,4} without overlapping should be very difficult. But fixed positions and orientations of vertecies may help a lot - if "centers" of cells will be movable (it’s a joke, of course: these centers are ideal points as well).

Too many things to do and absolutely no time for them… I want hyperbolic metrics of the time!

Good luck!

Andrey

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

>

> I was also thinking some yesterday about this thread between Andrey and

> Nan. Puzzles with infinite-sided cells seemed understandable (now that

> Andrey has shown the way!), but I had been having trouble wrapping my head

> around puzzles with infinite-sided vertex figures. So although I could see

> the {infinity,3} puzzle Andrey suggested would be a nice addition to

> MagicTile (even if the programming might be difficult), thinking about

> {3,infinity} was confounding. How would *that* behave?

>

> I went back to Don’s pictures that Melinda emailed about, and looked at the

> two views he has of the {3,infinity} tiling (the dual to the {infinity,3}).

> What is interesting is that the cells are "ideal triangles" - all the

> cell vertices lie at infinity, on the boundary of the disk. They are

> infinite in extent *in multiple directions* (though not infinite in area).

> So then I thought about how one would twist one of these cells, and

> concluded that it would not be possible to (isometrically) twist a cell

> without it overlapping other parts of the puzzle, since the vertices will

> remain on the disk boundary through the motion. So if you want to have a

> puzzle with an infinite vertex figure, you’ll be forced to have it behave in

> an impossiball-like fashion (at least for 2D puzzles).

>

> Furthermore, the slicing becomes degenerate for a {3,infinity} puzzle unless

> you want make some modifications to the previous MagicTile generalization.

> If you stick to circle slices (or sphere slices in H3), the only slicing

> circle encompassing the entire cell is the disk boundary itself, and a twist

> using that would be a rotation of the entire space. It would not permute

> anything. One option would be to have slicing circles which did not contain

> the entire cell though, and this could lead to some interesting puzzles.

> That seems like the only good way forward to me actually.

>

> I think the {4,4,4} situation will have some similarities to these

> thoughts. Each cell in the {6,3,3} tiling only has one ideal point at

> infinity, but the cells in the {4,4,4} tiling will have an infinite number

> of ideal points. It seems the result is that any twist (which moves all the

> material in a cell) will lead to overlapping material in the puzzle (?, I’m

> not 100%). The slicing certainly suffers from the same issues described

> above. It is the normal problems of puzzles without simplex vertex figures

> multiplied by a gazillion!

>

> Well, I could probably spew more, but I’ll stop in the hope that this

> doesn’t all just sound like gibberish. I realize that if one

> didn’t have some familiarity with the disk/ball models of hyperbolic spaces,

> it probably would. (The first six chapters of Visual Complex

> Analysis<http://www.amazon.com/gp/product/0198534469?ie=UTF8&tag=gravit-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0198534469>

> are

> awesome for learning about these models btw.)

>

> Roice

>