Thread: "n-torus"

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Did you see the Dual Circle
puzzle Melinda linked to the other day?

MagicTile supports many torus puzzles, including 4-colored puzzles with
square faces. However, it doesn't yet support the cuts in your picture.
Those are systolic cuts, similar to the cuts in the Earthquake puzzle we've
been discussing recently. Current MagicTile puzzles look more like this
, with circular cuts that could be
shrunk to a point on the surface. And of course MagicTile's puzzles aren't
physical.

I like your idea to go up a dimension too. You could have spherical
(rather than circular) cuts, or you could have systolic plane cuts there as
well, in which case it could be made to look somewhat like MagicCube4D,
with cubical faces cut up into 27 stickers each. I'm guessing you need at
least 8 colors in this case, giving 216 total stickers like you said.

I think the complexity of the commutators will grow for the torus just like
they do for the cube, roughly doubling in size with each increasing
dimension. And the possible piece types (1C, 2C, 3C, 4C, ...) will grow
similarly too.

Roice

P.S. The MagicTile abstraction considers the puzzle you pictured 2D, like
mathematicians consider a sphere 2D rather than 3D. Labeling your image a
3D puzzle is ok when clear, and I think I understood what you were meaning,
but if you want to embed it nicely as a flat torus it is also arguably 4D
. This is why I just like
calling it 2D :)


On Sat, Aug 20, 2016 at 10:20 PM, llamaonacid@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Has anyone seen a physical torus puzzle? I would like to see one in 3D an=
d
> higher dimensions. In 3D the minimum number of colors is 4 and there has =
to
> be some sticker-less surface if you wrap the stickers on the surface of t=
he
> torus. In the image you can add more columns or rows if you want. The 2D
> stickers does not look complex (you probably just need 2 algorithms to
> solve) but I think adding higher dimensions would be fun. The n-torus in
> mind would have 6^(n-1) stickers. My question is how complex would the
> algorithms or commutators for the n-torus be. Also, I would like to know
> the number of pieces it would have and compare it to the n-dimensional
> Rubik's Cube.
>
> -Guderian or Gude for short
>
>
>
>
>=20
>

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On 8/22/2016 12:32 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:
>
>> I hadn't thought through the face-centered earthquake case yet,
>> but using Arnaud's applet
>>
>> I just convinced myself there is no possible earthquake
>> face-twist, at least not based on systoles. Click the "systolic
>> pants decomp" option there and look at, say, the white "pair of
>> pants" in the center. A twist will move material within that
>> pair of pants, but at the end of the twist the new location of
>> all the shuffled material will need to cover the same original
>> area. If you pan certain heptagon vertices or edges to the
>> center of the view, you can see that this works. If you move a
>> heptagon center to the center of the view, it doesn't - there's
>> no way to make a 1/7th turn and get the pants to return to
>> covering the original area.
>
> OK, well now I'm having an out-of-pants experience but it made me
> realize that I didn't previously make myself clear. Once I clarify
> myself, you may conclude that I indeed have gone off into the
> weeds regarding catacombs. When talking about the "meta" puzzle, I
> was referring to the topology of the surface itself as opposed to
> the puzzle within it though maybe they're always identical. In the
> case of KQ, the genus is 3 making the topology that of a
> ball-and-stick model of a tetrahedron. When I talk about vertex
> twisting at the meta level I'm talking about cuts through the arms
> of this tetrahedron
> . So your
> current twists appear to cut three arms, twist one of them in
> place by 180 degrees while swapping the other two. What I called
> the pure "vertex" twist would sever three arms that meet at a
> meta-vertex, rotate that whole unit 120 degrees and reattach them
> all. The "edge" twist cuts 4 arms straight through the center of
> the tetrahedron and rotates one half by 180 degrees. By now you
> probably understand what I mean about "face" twists, which in the
> case of KQ is identical to a pure vertex twist opposite a given
> systolic triangle.
>
>
>
> I think we have been picturing things really similarly. I was also
> thinking of how the vertex and edge earthquake twists affected that
> thickened "tetrahedron", just like you. If you map the KQ surface to
> that tetrahedron such that a heptagon vertex maps to a tetrahedron
> vertex, it does work identically. And similarly for mapping a
> heptagon edge to a tetrahedron edge.
>
> I hadn't understood what you meant by a face being opposite a vertex
> before, but now I get that (on the KQ surface itself, each vertex has
> one opposite vertex and each face has two opposite faces). There is
> (of course) no way to map a heptagon center to a tetrahedron face,
> since the latter is not part of the surface, so the equivalence
> between this topology perspective and the surface perspective breaks down.
>
> Here's some more intuition as to why the equivalence breaks down in
> the face-centered case. The KQ surface has a lot more symmetries than
> a tetrahedron, but the symmetry group of the tetrahedron is a subgroup
> of the symmetry group of the KQ. A symmetry that rotates 1/7th a turn
> about a heptagon center is one of the KQ symmetries that is not a
> symmetry of the tetrahedral subgroup, whereas the vertex-centered and
> edge-centered twists are symmetries of both. Maybe this is also the
> reason why you can't have a face-centered earthquake twist (on the
> surface), because the systoles are arranged with tetrahedral
> symmetry. I bet I'm being clear as mud, but hopefully this adds
> something.

We may be talking past each other, but when I talk about "face
centered", I'm not talking about heptagons, but rather the four
triangular faces of the thickened tetrahedron. Since it's a tetrahedron,
those face twists are equivalent to the symmetric earthquake twist of
the opposite "thickened vertex". Talking about twists on this
tetrahedron (AKA "earthquake" twists) is at the "meta" level that I've
mentioned. I'm probably just adding more mud but maybe it will help.

-Melinda

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On 8/22/2016 12:32 PM, Roice Nelson d" href=3D"mailto:roice3@gmail.com">roice3@gmail.com [4D_Cubing]
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