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 Post subject: A Rigorous Distinction Between Opposite Face and Slice TurnsPosted: Sun Oct 06, 2013 9:49 am

Joined: Mon Sep 23, 2013 6:34 pm
There has been a great deal of discussion on the distinction between opposite face turns and center slice turns. Because most analysis in this department has used puzzles that function as as simulacra of a 3x3x3 (their group is a subset of the Rubik's Cube group), that is what I will focus on. All statements listed below apply only to the aforementioned simulacra. I also exclude any cubes that have unidirectional moves, a prime example of this is the Irreversible Cube.
At first, it seems that L', R is no different from M, x. On a typical Rubik's puzzle this is the case. However, many puzzles have mechanical constraints that cause a distinction to exist between face other than the color of a center piece. For an example, I suggest you investigate Oskar Van Deventer and Bram Cohen's Incomprehensible Cube(s). As one can see, since the centers have a fixed identification, the rotation of the cube afforded by the "x" commutes the fixed mechanical components. Therefore, the difference between an opposite slice turn and a middle slice turn in one such puzzle is a commutation of mechanics and nothing else.
A standard 3x3x3 has no distinguishable mechanics in different faces. For those not well versed in geometry, a face-transitive solid in one where, in essence, there is no way to distinguish the faces. A perfect cube is this, because you can flip and rotate the cube to replace any face with any other face without changing the appearance. If you are well versed in Geometry, it is something for which all faces are contained within the same symmetry orbit. So where am I going with this? I define a puzzle as mechanically face-transitive is each face has the same underlying mechanical structure and each face is inherently indistinguishable. Neither Incomprehensible Cube is face-transitive. Example of face-transitive puzzles include: A simple NxNxN cube, a Dino Cube, A Megaminx, a Dogic, and a Pyraminx. It does not include a Constrained Cube Ultimate, Belt Cube, or Enabler Cube. A puzzle which can become face-intransitive at any point is face-intransitive. As you can see, a puzzle does not have to be face-turning to be face-transitive.
Now, the tentative definition I've been building up to, for those hardy souls still reading this verbose post:
A middle slice turn is indistinct from an opposite face turn if the puzzle in question is mechanically face-transitive.
My original formulation contained an "if and only if" but I realized that one could make an M-turn if, say, the U and D faces were incapable of turning.
I encourage anybody to rip any portion of this to shreds; my goal is only to reach a fuller idea of various commutators in twisty puzzle groups by this definition.
Edit: Some errors of an embarrassingly trivial nature mentioned in below comments have been fixed.

Last edited by Topological Space on Sun Oct 06, 2013 7:22 pm, edited 5 times in total.

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 11:34 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
WOW!!! You joined Sep 23 and this is your first post? This HAS to qualify for the deepest first post I've seen yet on Twistypuzzles. No testing the waters first... just jump of the deep end. I like it. LOL!!! If you've been around a bit lurking you may know I love twisty puzzle theory and trying to hammer good definitions and the like so let me welcome you to the forums.
Topological Space wrote:
A middle slice turn is distinct from an opposite face turn if and only if the puzzle in question is mechanically face-transitive.
My original formulation contained an "if and only if" but I realized that one could make an M-turn if, say, the U and D faces were incapable of turning.
I encourage anybody to rip any portion of this to shreds; my goal is only to reach a fuller idea of various commutators in twisty puzzle groups by this definition.
Isn't that "if and only if" still in there or am I missing something. Also you may want to check out this table I made for my Multi Gear Cube Kit.
http://wwwmwww.com/Puzzle/Oskar/GearPermCorrection.png
Items 4, 18, 27, 32, and 34 on this list are cases where middle slice turn is NOT distinct from an opposite face turn yet only puzzle 32 meets your definition of mechanically face-transitive I believe. Oh and reading your claim "A middle slice turn is distinct from an opposite face turn if and only if the puzzle in question is mechanically face-transitive.".... shouldn't there be a NOT in there somewhere?

I think I would agree with:

A middle slice turn is distinct from an opposite face turn if and only if the puzzle in question is NOT mechanically face-transitive.

Either that or I'm misunderstanding something.

Again welcome to the forums,
Carl

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 12:25 pm

Joined: Thu Dec 15, 2011 10:04 pm
The latest WCA Notation omits single-slice moves such as “M”: WCA Notation. Previously "M" was defined as a turn of the middle-slice layer, following the same direction as L.

Here are before and after shots of two 3x3x3 puzzles:
Attachment:

L' R vs M x Comp_sm.jpg [ 174.29 KiB | Viewed 655 times ]
The sequence (L’ R) was applied to one and (M x) was applied to the other. Is there any way to distinguish which puzzle had which sequence applied? I would say that the sequences are equivalent if the final state of the puzzle is identical.

Here is the Quarter Cube:
Attachment:

Quarter-Cube-no-L' R or M x_sm.jpg [ 109.42 KiB | Viewed 655 times ]
I believe that it IS mechanically face-transitive, at least in the initial state. The faces of this puzzle are initially constrained to 90° clockwise turns. The sequence (L’ R) is thus not possible because L’ is a counter clockwise turn. Interestingly, the sequence (M x) is also not possible. In fact, no middle slice turns are available - M, E or S - in any direction. The only way to include the M layer in a turn is to do a wing turn. For example, on the puzzle as shown, Rw is available. This makes me wonder if there are any mechanisms that would allow one but not the other of (L’ R) and (M x).

Finally, the 3x5x7 with (L’ R) and (M x) applied:
Attachment:

3x5x7_L'-R-vs-M-x_Comp_sm.jpg [ 101.5 KiB | Viewed 655 times ]
Not a cubic puzzle, not a 3x3x3 puzzle and a reminder of what “M” means. Perhaps we need additional notation to refer to “all the layers between the outermost layers” such as (M*).

Quote:
A middle slice turn is distinct from an opposite face turn if and only if the puzzle in question is mechanically face-transitive... I define a puzzle as mechanically face-transitive is each face has the same underlying mechanical structure and each face is inherently indistinguishable.
Is your argument that for mechanically face-transitive puzzles (L' R) and (M x) are equivalent and thus NOT distinct? In any case, can we say anything about puzzles that are not mechanically face-transitive? For example, will we find a puzzle that is not mechanically face-transitive where (L' R) and (M x) are equivalent?

It is an interesting idea and for the record, I'd love to see a cubic puzzle where "x" (or y or z) changes the behaviour of a puzzle (i.e. an orientation-sensitive latch cube).

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 1:24 pm

Joined: Mon Sep 23, 2013 6:34 pm
wwwmwww wrote:
Also you may want to check out this table I made for my Multi Gear Cube Kit.
http://wwwmwww.com/Puzzle/Oskar/GearPermCorrection.png
Items 4, 18, 27, 32, and 34 on this list are cases where middle slice turn is NOT distinct from an opposite face turn yet only puzzle 32 meets your definition of mechanically face-transitive I believe.

I cannot speak to most of the other cubes listed, as my knowledge of gear cubes is paltry. But I believe that the slice gear cube is not face-transitive because the 3rd plane of movement works differently that the first 2, and as such the faces parallel to the 3rd plane work differently. I am ashamed to say I own no geared puzzles and do not fully understand how a middle slice turn works on a geared cube; I will stop that line of investigation before I make a fool of myself.
On the subject of face-transitivity, I agree with your assessment that #32 is face-transitive. However, there is broader scope for face-transitive geared puzzles, namely all those (including #32) with identical gearing ratios in all 3 dimensions. That would include #16 and #26, possibly #35 although I would have to conduct a more thorough investigation to determine whether a 1:4 1:4 1:4 puzzle is truly equivalent to a Fused Cube

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 1:37 pm

Joined: Mon Sep 23, 2013 6:34 pm
Pete the Geek wrote:

I believe that [the Quarter Cube] IS mechanically face-transitive, at least in the initial state. The faces of this puzzle are initially constrained to 90° clockwise turns. The sequence (L’ R) is thus not possible because L’ is a counter clockwise turn. Interestingly, the sequence (M x) is also not possible. In fact, no middle slice turns are available - M, E or S - in any direction. The only way to include the M layer in a turn is to do a wing turn. For example, on the puzzle as shown, Rw is available. This makes me wonder if there are any mechanisms that would allow one but not the other of (L’ R) and (M x).

I am currently wrestling with the question as to whether a puzzle that is face-transitive in the solved state but can become face-intransitive is truly face-transitive, because after all, the solved state is an arbitrary selection from the set of possible states. If I decide that it is face-transitive, then the question below is moot (note that I am only dealing with 2-way puzzles: any turn can be undone. The Irreversible Cube is NOT an example of this, but most puzzles are).
In the concept of face-transitivity in mathematics, the symmetry mappings may include reflection or rotational symmetry. Because there is no precedent in Geometry for transformations such as those experienced by a twisty puzzle, I am unsure whether I should include turns of the puzzle in the set of transformational operations allowed for symmetry mappings. Again, this and the above question are linked. Should somebody other than I provide a mathematical reason one way or another, I will likely settle on that definition.

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 2:32 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Full face-transitivity isn't necessary for slice and opposite face moves to be equivalent. All you need is for the puzzle to be rotationally symmetric along the axis being sliced. There are some funny other cases where they're equivalent though, for example in 3x3x3 where each axis is either slice or opposite face turning, it doesn't matter which is which.

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 3:39 pm

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Pete the Geek wrote:
The latest WCA Notation omits single-slice moves such as “M”: WCA Notation. Previously "M" was defined as a turn of the middle-slice layer, following the same direction as L.

Here are before and after shots of two 3x3x3 puzzles:
Attachment:
L' R vs M x Comp_sm.jpg
The sequence (L’ R) was applied to one and (M x) was applied to the other. ....
Pete, I guess you wanted to write (M x'). An M slice turn followed by a 90 ° counter clockwise rotation of the whole cube around the x axis.

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 Post subject: Re: A Rigorous Distinction Between Opposite Face and Slice TPosted: Sun Oct 06, 2013 3:53 pm

Joined: Mon Sep 23, 2013 6:34 pm
Bram wrote:
Full face-transitivity isn't necessary for slice and opposite face moves to be equivalent. All you need is for the puzzle to be rotationally symmetric along the axis being sliced.

That is exactly why it is an "if" statement rather than the untrue "if and only if" I used earlier. I will keep your symmetry comment in mind, as I had not discovered that.

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