   Date: Thu, 24 Feb 83 18:15:00 -0500 (EST)   From: Dan Hoey <Hoey@CMU-CS-A >
~~~ ~~~ Subject: Whole-Cubing

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<Winning Ways>, 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
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.     