Date: 24 February 1983 1815-EST (Thursday) From: Dan Hoey at CMU-CS-A To: Cube-Lovers at mit-mc Subject: Whole-Cubing Message-Id: <24Feb83 181536 DH51@CMU-CS-A> When we describe cube positions, we typically fix the position of the face centers. This avoids counting different positions of the same pattern more than once. But suppose we wished to distinguish between different positions of a pattern? How should we form this group G* of spatially oriented patterns? A simple way of generating G* is to adjoin generators for C, the motion group of the cube, to generators for G, the usual Rubik group. Generators for C were discussed in early Cube-Lovers mail as I, J, and K, three orthogonal quarter-twists of the whole cube in space, although there was some disagreement about which was which. We generate G with B, F, U, D, L, and R as usual. Unfortunately, the two kinds of generators are not conjugates, which was a nice thing about our generators for G. Also the generators do not interact very strongly. If we have an identity on {IJKBFUDLR}, the substring on {IJK} must be an identity in C, and there is a simple way of transforming the substring on {BFUDLR} to be an identity in G. In @i, Berlekamp, Conway, and Guy described another way of generating G*, by appending the slice moves, which they name by mnemonic greek letters; this labeling scheme was also reported in Hofstadter's column. Here we have some stronger interaction between the two kinds of generators, but they are still two different kinds: they are not conjugate. This has never really bothered the English, though; they gratuitously include half-twists as generators as well. I thought up a scheme involving what I will call depth-2 moves named B2, F2, U2, D2, L2, and R2. Readers of my reports on the 4^3 and 5^3 cubes will find these familiar. Essentially, a depth-2 move is performed by turning a face together with its adjacent center slice. Thus F2 involves holding the B face immobile and turning the rest of the cube a quarter-turn clockwise, as seen from the front. Clearly the depth-2 moves are M-conjugate. Unfortunately, they do not generate all of G*, nor even all of G. This can be seen easily by noticing that each depth-2 move is an even permutation of the edge cubies, while G includes odd permutations of edges. We can look at this a different way by observing that a depth-2 move is the same as a vanilla depth-1 move of the opposite face together with a whole-cube quarter-twist. Thus we can perform any depth-1 process using depth-2 moves, and observe the spatial orientation of the cube at the end. If we performed an odd number of depth-2 moves, then there were an odd number of whole-cube moves, so the cube cannot be in its home orientation at the end. So just what group do the depth-2 moves generate? It turns out that they generate precisely half of G*, the half that contains even edge permutations. Suppose we want to produce such a position in G*. First operate in G, simulating depth-1 moves as described above, to produce a position that is an even number of whole-cube moves from the desired one. Then use M-conjugates of the process F2 L2 D2' B2' D2' F2 L2 B2 L2 U2' F2' D2' R2' B2, which performs a whole-cube third-twist. [This was derived from identity I14-4]. Since all the even whole-cube moves are generated from third-twists, this is all you need. What can we do about generating all of G* with a set of conjugate generators? Sad to say, that is impossible. For supposing we had such generators, their cycle structures would have to be the same; in particular, they would have to have the same permutation parities. Applying an odd number of such generators would yield those parities, and applying an even number would yield even parity on every kind of cubie. But G* includes four different parity classes: Face centers Edges Corners even even even even odd odd odd even odd odd odd even so at most half of G* can be generated with a set of conjugate generators.