Message #1045

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Magic Tile again
Date: Sun, 18 Jul 2010 16:56:50 -0500

Thanks for making these intriguing observations Andrey! I had noticed the
strange fact on the octagonal 6-color puzzle that visually opposite stickers
in a given cell represented the same "logical" sticker (so not only were the
same logical faces being drawn multiple times in the hyperbolic plane, but
the same logical stickers were drawn multiple times in a given cell). I
never actually took the time to count how many logical stickers were on a
face though, so was surprised to find out it is 9, and that the
combinatorial behavior is same as the classic puzzle. Wow!

I’ve played around a little with the non-orientable spherical puzzles (a
3-colored cube and a 6-colored dodecahedron, having opposite faces of the
polyhedra identified), and so it is interesting that the 3-colored puzzles
are combinatorially the same as the former. The animations will be
different in how I planned to do the non-orientable version, since I
imagined some faces would rotate in a counter fashion to others during a
twist. But sure enough, the end result of the twists will be the same. I
think the 6-colored dodecahedron will be a mathematically new puzzle.

As for the potential new puzzles you mention, the low genus classification
work <http://tilings.org/#lowgenus> at Tilings.org can help answer some
questions about what is possible, specifically this
table<http://tilings.org/pubs/tileclasstables.pdf>.
The second and third columns are relevant, the second column giving the
order of the rotation group they are listing, and the third column the
tiling symbol. MagicTile currently looks only at tilings of the form
(2,3,p) in their notation. If you take the group order and divide by p,
this will tell you the number of colors there would be for the corresponding
puzzle. So I can see that an 18-colored {12,3} is indeed possible because
there is a genus 10 surface in the list having a 12-gon tiling and a
rotation group with order 12*18=216. There is no listing of a nonagonal
tiling with order 9*12=108, though I don’t think this guarantees it is
impossible since this list is limited to surfaces of genus 13 or less. I am
curious to hear more of why you suspect this nonagonal possibility (did you
also suspect an 8-color "double pyraminx" as possible? why or why not?).
There is an entry in the table implying there could be a 36 color nonagonal
puzzle.

From the size of the table, one can see that there a *huge* number of
potential puzzles that are possible if one removes the restriction of 3
polygons meeting at a vertex.

I’ll close with something unique about the 12 colored octagonal, which may
or may not lead to further insight (I don’t know if it is significant). But
it is the only hyperbolic puzzle in the MagicTile list where copied cells
are generated by an odd number of reflections (3 reflections). Find a cell,
and check the shortest path to one of its repeats to see what I mean, and
compare this with other puzzles.

Cheers,
Roice

On Sat, Jul 17, 2010 at 11:19 AM, Andrey <andreyastrelin@yahoo.com> wrote:

> I’ve took a closer look to Magic Tile set of puzzles and found a strange
> thing.
> Of 11 puzzles from "hyperbolic" part of set there are only 5
> mathematically different ones and two of them are already listed in
> "spherical" section: all 6-colors are equivalent to Rubik’s cube, all
> 4-colors are alternative implementations of pyraminx, and 3-colors are
> equivalent to 3-colored Rubik’s cube - non-oriented polyhedron with one
> vertex, 3 edges and 3 digonal faces :) (are they the same as "digonal"
> puzzle? No, there is not enough 1C pieces in the latter). Two others -
> 24-color Klein’s quartic and 12-colors {8,3} puzzle (double cube?) are
> really hyperbolic. This {8,3} looks very interesting - and I have to
> understand how it works. Like we cut a hole in center of paper cube,
> dupicated the rest and interconnented copies so that when you pass some of
> edges you go to another cube… May be not. And it looks like there could be
> 12-colored {9,3} (triple pyraminx) and 18-colored {12,3} - triple cube or
> double {6,3}*9 colors.
> 3 colors, 7 layers took some time to understand and solve it - most
> algorithms from N^3 didn’t work. Luckily there is only small set of colors
> distribution, and simplest commutators did the trick :)))
> Thanks again, Roice!
>
> Andrey
>
>