4x4x4 centers permutation (cycles)?

[bcube]

Hi,

on a 3x3x3 cube one quarter move of outer layer creates 1 4-cycle of corners (odd permutation) and 1 4-cycle of edges (odd permutation).

Now, I'm wondering if it applies too for a 4x4x4 cube too (not a 4x4x4 supercube). At first sight it seems one quarter move of outer layer creates (among other) 1 4-cycle of corners (odd permutation) and 1 4-cycle of centers (odd permutation). If we look at the centers orbit alone, it certainty can not be talked about (either odd or even) permutation at all, because the term permutation requires the centers to be distinguishable. Can it be said at least that the centers orbit is parity-dependent on the corners orbit, similarly to the edges orbit and corners orbit on a 3x3x3?

If not, is it correct to say that on quarter move of outer layer on a 4x4x4 (not 4x4x4 supercube) creates 1 4-cycle of corners and 1 4-cycle of wings only, because the cycle of centers is not defined? Similarly one quarter move of inner layer creates only 1 4-cycle of wings because the cycles of the centers are not defined?

 

[Goosly]

because the term permutation requires the centers to be distinguishable

Is that a fact, or you opinion?

 

[cuBerBruce]

Is that a fact, or you opinion?

Well, it seems the word permutation is often generalized/extended to allow indistinguishable objects. But if we're talking about permutation cycles and even and odd permutations, then I would say we must be talking about the more strict definition of permutation.

As to bcube's question, I would say that we have to be careful whether we're talking about how pieces get moved around by a maneuver, or if we simply looking at visibly distinguishable positions. For instance, the typical "dedge flip" maneuver not only swaps two edge pieces, but also does two swaps of center pieces on the U face. If we try to create a UFl-DBr swap algorithm by using 2R' (SiGN notation) or r' (Singmaster notation) as a setup move to this "dedge flip" algorithm, then we find that our UFl-DBr swap maneuver messes up centers. This is because the "dedge flip" maneuver we used was not supercube safe. This is to illustrate how even though we are not talking about a supercube, how we may still get messed up by ignoring how maneuvers actually move pieces around.

 

[bcube]

cuBerBruce, thank you for your reply. I'm obviously not as strong at deducing as you / others are. Therefore, unless someone else do it, I would like to ask you if you could answer more explicitly, please? Ideally by yes or no .

I am interested in these questions (for 4x4x4 cube):

1) Can it be said at least that the centers orbit is parity-dependent on the corners orbit, similarly to the edges orbit and corners orbit on a 3x3x3?

2) If not, is it correct to say that one quarter move of outer layer creates 1 4-cycle of corners and 1 4-cycle of wings only, because the cycle of centers is not defined (however, for 4x4x4 supercube it is defined)?

 

[qqwref]

1) Can it be said at least that the centers orbit is parity-dependent on the corners orbit, similarly to the edges orbit and corners orbit on a 3x3x3?

Yes - an outer layer turn changes the corner and center parity, and an inner layer turn changes the edge parity only. Of course, on a normal 4x4x4 the center orbit parity cannot be seen.

2) [...] is it correct to say that one quarter move of outer layer creates 1 4-cycle of corners and 1 4-cycle of wings only, because the cycle of centers is not defined (however, for 4x4x4 supercube it is defined)?

No, because the cycle of centers is defined - there are just sets of identical centers. On a solved cube an outer layer turn does not seem to affect the centers, but that is only because those four centers happen to all be identical. If the cube was scrambled, you would see it doing the 4-cycle of centers. The move always has the same effect, but because of identical pieces, sometimes the ending state and starting state of the centers are the same.

 

[bcube]

No, because the cycle of centers is defined - there are just sets of identical centers. On a solved cube an outer layer turn does not seem to affect the centers, but that is only because those four centers happen to all be identical. If the cube was scrambled, you would see it doing the 4-cycle of centers. The move always has the same effect, but because of identical pieces, sometimes the ending state and starting state of the centers are the same.

I understand a permutation as a bijective map (not sure if that term in English is what I mean), thus identical pieces are not allowed in that model.

You say that the cycle of centers is defined. Imagine I want to determine the permutation of centers after each move. I can look at the corners and assign the same (odd or even) permutation to the centers. But if I manually (i.e. not by legal turns) swap two center-pieces (thus changing their permutation), the cube would be still solvable, even if the permutation of corners and the permutation of centers wouldn't be the same anymore (which is not possible on 4x4x4 supercube ).

So, is there an unambiguously way how to determine a permutation of centers (for example in solved state)? How is that unambiguously way defined? I understand your point about scrambled cube though, where the centers are treated as not identical (supercube style).

 

[bcube]

So, if I am operating on bijective map definition of permutation (wiki says it is what I meant), could anyone confirm that the answer to my question 1 is no and answer to my question 2 is yes, please?

Proving me wrong on either my understanding of permutation or answers to my questions is welcome, too.