Thread: "Magic Tile again"

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



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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 at Tilings.org can help answer some
questions about what is possible, specifically this
table.
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 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
>
>

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Thanks for making these intriguing observations Andrey! =A0I had noticed th=
e strange fact on the octagonal 6-color puzzle that visually opposite stick=
ers in a given cell represented the same "logical" sticker (so no=
t only were the same logical faces being drawn multiple times in the hyperb=
olic plane, but the same logical stickers were drawn multiple times in a gi=
ven cell). =A0I never actually took the time to count how many logical stic=
kers 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. =A0Wow!



I've played around a little with the non-orientable sphe=
rical puzzles (a 3-colored cube and a 6-colored dodecahedron, having opposi=
te faces of the polyhedra identified), and so it is interesting that the 3-=
colored puzzles are combinatorially the same as the former. =A0The animatio=
ns will be different in how I planned to do the non-orientable version, sin=
ce I imagined some faces would rotate in a counter fashion to others during=
a twist. =A0But sure enough, the end result of the twists will be the same=
. =A0I think the 6-colored dodecahedron will be a mathematically new puzzle=
.


As for the potential new puzzles you mention, the ef=3D"http://tilings.org/#lowgenus">low genus classification work at Ti=
lings.org can help answer some questions about what is possible, specifical=
ly this table. =
=A0The second and third columns are relevant, the second column giving the =
order of the rotation group they are listing, and the third column the tili=
ng symbol. =A0MagicTile currently looks only at tilings of the form (2,3,p)=
in their notation. =A0If you take the group order and divide by p, this wi=
ll tell you the number of colors there would be for the corresponding puzzl=
e. =A0So I can see that an 18-colored {12,3} is indeed possible because the=
re is a genus 10 surface in the list having a 12-gon tiling and a rotation =
group with order 12*18=3D216. =A0There is no listing of a nonagonal tiling =
with order 9*12=3D108, though I don't think this guarantees it is impos=
sible since this list is limited to surfaces of genus 13 or less. =A0I am c=
urious to hear more of why you suspect this nonagonal possibility (did you =
also suspect an 8-color "double pyraminx" as possible? =A0why or =
why not?). =A0There is an entry in the table implying there could be a 36 c=
olor nonagonal puzzle.


From the size of the table, one can see that there a *h=
uge* number of potential puzzles that are possible if one removes the restr=
iction of 3 polygons meeting at a vertex.

I'll=
close with something unique about the 12 colored octagonal, which may or m=
ay not lead to further insight (I don't know if it is significant). =A0=
But it is the only hyperbolic puzzle in the MagicTile list where copied cel=
ls are generated by an odd number of reflections (3 reflections). =A0Find 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


ss=3D"gmail_quote">On Sat, Jul 17, 2010 at 11:19 AM, Andrey r"><andreyastrelin@yahoo.com=
>
wrote:

x #ccc solid;padding-left:1ex;">I've took a closer look to Magic Tile s=
et of puzzles and found a strange thing.

=A0Of 11 puzzles from "hyperbolic" part of set there are only 5 =
mathematically different ones and two of them are already listed in "s=
pherical" section: all 6-colors are equivalent to Rubik's cube, al=
l 4-colors are alternative implementations of pyraminx, and 3-colors are eq=
uivalent to 3-colored Rubik's cube - non-oriented polyhedron with one v=
ertex, 3 edges and 3 digonal faces :) (are they the same as "digonal&q=
uot; 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 under=
stand how it works. Like we cut a hole in center of paper cube, dupicated t=
he rest and interconnented copies so that when you pass some of edges you g=
o to another cube... May be not. And it looks like there could be 12-colore=
d {9,3} (triple pyraminx) and 18-colored {12,3} - triple cube or double {6,=
3}*9 colors.


=A03 colors, 7 layers took some time to understand and solve it - most alg=
orithms from N^3 didn't work. Luckily there is only small set of colors=
distribution, and simplest commutators did the trick :)))

=A0Thanks again, Roice!



=A0Andrey





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From: "Andrey" <andreyastrelin@yahoo.com>
Date: Tue, 20 Jul 2010 18:59:22 -0000
Subject: Re: [MC4D] Magic Tile again



Roice,
I've missed the "double pyraminx" because it should give "planar" tile - =
{6,3}, 8 colors. But now I see that it's an excellent opportunity to check =
my theory - and this theory doesn't work! We cannot "upgrade" 4-colored {6,=
3} to 8 colors, but easily design 12-colored tile. So it's some "triple pyr=
aminx" but with only double size of face.
I'm investigating {9,3} patterns now. And sometimes I fill that there exi=
sts 12-color tile (and 16-color) and sometimes not. It's difficult to under=
stand hyperbolic transitions of plane, but very interesting...

Andrey

>I am
> curious to hear more of why you suspect this nonagonal possibility (did y=
ou
> 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 nonago=
nal
> puzzle.
>=20
> 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.
>=20
> I'll close with something unique about the 12 colored octagonal, which ma=
y
> 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 ce=
ll,
> and check the shortest path to one of its repeats to see what I mean, and
> compare this with other puzzles.
>=20
> Cheers,
> Roice
>=20




From: "matthewsheerin" <damienturtle@hotmail.co.uk>
Date: Tue, 20 Jul 2010 21:55:52 -0000
Subject: Re: Magic Tile again



I know little about hyperbolic geometry, and didn't realise how the Magic T=
ile puzzles could correlate to familiar puzzles. Just to point out a sligh=
t technical issue: the four colour puzzles don't seem to quite match the py=
raminx, but rather this puzzle http://users.skynet.be/gelatinbrain/Applets/=
Magic%20Polyhedra/tetra_fv1.htm.

Matt

--- In 4D_Cubing@yahoogroups.com, "Andrey" 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 mathematic=
ally different ones and two of them are already listed in "spherical" secti=
on: all 6-colors are equivalent to Rubik's cube, all 4-colors are alternati=
ve implementations of pyraminx





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