Message #3504

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] n-torus
Date: Mon, 22 Aug 2016 15:06:43 -0500

Did you see the Dual Circle <https://www.youtube.com/watch?v=uOAwjIebYDA>
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
<https://goo.gl/photos/LDwU13TPJqLGRrwN6>, 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
<https://en.wikipedia.org/wiki/Clifford_torus>. 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 and
> 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 the
> 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
>
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