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Cross-Check Patterns

Submitted by B MacKenzie on Wed, 12/08/2010 - 13:44.

By applying the 24 rotation symmetries to the corner facelets of the cube one may generate the Cross Pretty Pattern Group. These patterns may be arranged into five conjugate classes: the identity cube, six order two 6-cross patterns, eight order 3 6-cross patterns, six order 4 4-cross patterns and three order 2 4-cross patterns.

By applying the 24 Th symmetries to the edge facelets of the cube one may generate the Check (or Checkerboard ) Pretty Pattern Group. These patterns may be arranged into six conjugate classes: the identity cube, pons asinorum, eight order three 6-check patterns, eight order six 6-check patterns, three order two 4-check patterns and three order two 2-check patterns.

The product of these two groups is the Cross-Check Pattern Group to coin a term. The 576 elements of this group have various combinations of plaid, cross, check and spot patterns on the cube faces. Symmetry reduces this group to 46 basic patterns (45 minus the identity cube). I have put together a table describing these patterns with turn sequences for their generation. Those interested in pretty patterns may find the table in the Rubik section of my web site.

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It is quite interesting that

Submitted by kociemba on Thu, 12/09/2010 - 12:36.
It is quite interesting that you also get exactly 46 symmetry reduced basic patterns with the Pattern Editor of Cube Explorer, if you allow the 5 involved face patterns on all 6 faces: clean, spot, check, cross, "plaid". So it is true, that the cubes of the Cross-Check Pattern Group are exactly those, where the faces are allowed to have any of these five patterns - which is not obvious at all.
All of these cubes also are antisymmetric, so that reduction by symmetry + inversion does not reduce the number of classes.
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Symmetry of Inverses

Submitted by Jerry Bryan on Fri, 12/10/2010 - 13:13.
I'm not sure if this is what you are talking about or not, but if we let Symm(x) be the set of symmetries m of x such that x=m-1xm (Symm(x) will be a subgroup of the symmetries M), then Symm(x)=Symm(x-1).

Jerry
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Yes, that is interesting. Th

Submitted by B MacKenzie on Thu, 12/09/2010 - 16:24.

Yes, that is interesting. There is probably some elegant argument showing that that must be the case, although I'm not mathematician enough to see it. In addition to antisymmetry, all the elements have some type of mirror symmetry. Which means one can make do without the mirrored conjugations. One can get to all the members of an equivalence class simply by rotating the cube and applying the representative turn sequence.

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cross-check patterns

Submitted by brac37 on Fri, 12/10/2010 - 14:46.
For cross-check patterns, the corners must match relatively and be solved with respect to each other. Hence the corners are rotated as a whole and can only be permuted in an even manner. This makes that the edges are even as well. Since the edges must correspond to each other, only a symmetry is possible to move them. Only Th-symmetries are even as permutations of the edges, thus only these symmetries are possible as such.

About the symmetry and antisymmetry, I do not see an elegant argument to prove their existence. I discovered a few months ago that positions of the corner square group have both a symmetry and an antisymmetry as well.
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Antisymmetric Cubes

Submitted by B MacKenzie on Fri, 12/10/2010 - 17:27.

I too am puzzled as to how one might frame an elegant criteria to indicate if a cube state is antisymmetric or not. The question is whether the symmetry equivalence class of an element includes the element's inverse. For example the three corner swap:

    R2 F2 R' B' R F2 R' B R'

has no normal symmetry yet is antisymmetric. A dihedral plane symmetry conjugation does the trick. On the other hand the three corner swap:

    R B R' F R B' R' F'

has no symmetry at all, normal or antisymmetric. Add a double flip and its antisymmetric:

    D' L D R' D' L' F L F' R' F L' F' R D R
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