Archives

I wanted to post a number of miscellaneous items about representing the cube, and I will also include a few other related items.

I'll start with the group S₃ as an example.  I will treat the group S₃ as acting on the set {0, 1, 2}.  As I have been doing recently, I'll use the notation (a b c) to represent the permutation 0→a, 1→b, 2→c. 

In this notation, the entirety of S₃ can be listed as follows:

(0 1 2)
(0 2 1)
(1 0 2)
(1 2 0)
(2 0 1)
(2 1 0)

This basic idea, or something very similar to it, is probably the way most people represent the cube in a computer program.  Variations on the theme could include an S₅₄ model, an S₄₈ model, an S₂₄ × S₂₄ model, and some sort of wreath product model.  The most common wreath product model would probably be something like (S₈ wr C₃) × (S₁₂ wr C₂).  In the wreath product model, S₈ and S₁₂ represent the corner cubies and the edge cubies, respectively, and C₃ and C₂ represent the twists of the corner cubies and the flips of the edge cubies, respectively.  Of course, none of these various models are isomorphic to the cube group.  Rather, the cube group is a subgroup of whatever group is chosen as the computer model.