On 01/08/95 at 13:47:00 Martin Schoenert said: >Jerry Bryan wrote in his e-mail message of 1995/01/07
Note that this model does not include the face centers. That is, it
is G[C,E] rather than G[C,E,F]. 56 numbers would be required to
include the face centers. The distinction between 48 facelets and
56 facelets bears on the nitpicky question of whether the set C of
rotations is a subgroup of G or not.
Absolutely right. This part of the GAP documentation was written years
ago. These days I represent MG, CG, etc. as permutation groups on 54
points. I also changed the numbering, so that the [1..24] represent the
edges, [25..48] points represent the edges, and [49..54] represent the
Martin was kind enough not to point out that 6*9 is 54 rather than 56.
When I write the model out to disk, I only write out 8 corner facelets
and 12 edge facelets. For example, I only write out the front and
back corner facelets. This saves space and converts the model from
a facelet model to a cubie model, with the twists implicitly encoded
rather than being explicitly encoded via multiplication tables. It
also automatically establishes a frame of reference by which a
proof of conservation of twist and flip can be accomplished.
In terms of computational group theory this sequence of 8 corner and
12 edgde facelets is called a *base* for the permutation group G.
That is, each element of the group is uniquely determined by the
images of those 20 facelets. Of course if you have already proved
that no single corner can be twisted and no single edge can be flipped,
you can reduce this to 7 corner and 11 edge facelets.
With some of my models it can be reduced slightly more. Since
representative elements of equivalence classes are what are being stored,
the representative element can be chosen to fix a cubie of your choice.
The cubie which is being fixed need not be stored. This trick does
not work, of course, if all cube positions are being stored rather
than just representative elements.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU