# Symmetries and Antisymmetries of Rubik's cube positions



## Herbert Kociemba (Oct 6, 2013)

I just finished the classification of all Rubik's cube concerning their symmetry/antisymmetry. If you are interested, have a look at
http://cubezzz.dyndns.org/drupal/?q=node/view/526


----------



## Christopher Mowla (Oct 6, 2013)

From doing my single commutator project, I have become very interested in symmetry reductions and other reductions to manage to only have 967 corner cases to solve with a single commutator instead of (8!/2)(3^7) (which I have completed) and currently 3407 cases for middle edges (but I am currently trying to prove yet another reduction that will handle take out an additional 1316 cases, and I hope to find more reductions after that) instead of (12!/2)(2^11) to test. (I'm coding software in Mathematica to help me make these calculations and generate cases so that I'm positive that I am not making any mistakes).

I have heard off and on about symmetries of the cube, but I have minimal knowledge of group theory and don't seem to posses the aptitude to grasp the abstract material at hand without some illustrative examples. From my commutator project, I believe I am actually practicing group theory without any prior knowledge, but it still bothers me that I have to teach myself these concepts from scratch as I need them instead of having the aptitude to benefit from the works of others before my time. Jaaps pages are not helpful for me either (neither is Wikipedia).

Would you be kind enough to give cube states (in maneuver form) which are symmetrical to each other to express to people like me exactly what you're doing in a more concrete form?


----------



## Herbert Kociemba (Oct 7, 2013)

cmowla said:


> Would you be kind enough to give cube states (in maneuver form) which are symmetrical to each other to express to people like me exactly what you're doing in a more concrete form?




First of all it is important not to confuse a symmetry of a cube in general with the symmetry of a specific Rubik's cube.
A cube in general has 24 ways of rotating it into different positions. The mirror images of these positions contribute another 24 cases. So the group M of cube symmetries has 48 elements, each one describing how to rotate/mirror a cube into the new position.
If you have a specific Rubik's cube x, each of these 48 elements can act on x. This is called the group action of M. Let's take the cube x for example be the cube with the generator U R. The 24 rotations of M acting on this cube give (including the do nothing rotation) the new cubes with the generators:

U R, D F, D B, F L, L D, B L, R U, U F, U L, U B, B U, D R, F R, R D, D L, L U, R F, F U, F D, L B, L F, B D, B U, R B

Try it. You can always fix two points of the U R cube with two fingers and rotate it into a new position for each of the generators given above.

For the mirror images I personally prefer the point reflection at the center of the cube. The point reflection sends U to D', U2 to D2 and U' to D. It sends R to L', R2 to L2 and R' to L......
Doing this for the 24 maneuvers given above we get:

D' L', U' B', U' F', B' R', R' U', F' R'........ 

All these 48 cubes look different, but essentially they are the same. The action of M on x generated 48 different cubes, these 48 cubes are called the orbit of x under the group action.

Not lets take another cube y = U.

If you apply the 48 elements of M to this cube, you only get

U, R, F, D, L, B, D', L', B', U', R', F'.

The orbit of y has only 12 different cubes. This is, because y itself has some symmetry. If you rotate y about the UD-axis, nothing will happen. There are 4 elements of M, which keep y fixed: UD-Axis do nothing rotation, UD-Axis 90 deg rotation, UD-Axis180 deg rotation and UD-Axis 270 deg rotation.
We say the cube y has the symmetry of the group C4, or the group C4 is the stabilizer group of the cube y.

There are are essentially 33 different subgroups of M and therefore 33 different kinds of symmetry a specific cube can have. You can find them on my homepage.


----------



## d2alphame (May 17, 2015)

This is the best explanation I have ever seen. Thanks Herbert.


----------

