[next] [prev] [up] Date: Sat, 07 Jan 95 10:15:52 -0500 (EST)
[next] [prev] [up] From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
[next] [prev] [up] Subject: Re: Cube with GAP
On 01/06/95 at 23:51:00 mark.longridge@canrem.com said:

As I understand it, the format Martin uses in GAP is to represent
the 3x3x3 cube by assigning each individual facelet an unique
number like so (by the way, the following part is all from the
GAP documentation).

----------------------------------------------------------------------
                      +--------------+
                      |  1    2    3 |
                      |  4  top    5 |
                      |  6    7    8 |
       +--------------+--------------+--------------+--------------+
       |  9   10   11 | 17   18   19 | 25   26   27 | 33   34   35 |
       | 12  left  13 | 20 front  21 | 28 right  29 | 36  rear  37 |
       | 14   15   16 | 22   23   24 | 30   31   32 | 38   39   40 |
       +--------------+--------------+--------------+--------------+
                      | 41   42   43 |
                      | 44 bottom 45 |
                      | 46   47   48 |
                      +--------------+

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. What I don't see is how to
model the Supergroup in GAP. It looks like you would have to
label each Face center with four numbers so you could see the
rotations of the Face centers, but that seems like overkill.

I call this kind of model a facelet model
rather than a cubie model, and the twists and flips are implicit in
a facelet model. I would think that the twists and flips would have
to be made explicit in a cubie model. Dan Hoey reported to me once
that he had an error wherein his corners turned themselves inside
out. I can't totally picture how that happened, but it was related
to the fact that he was using a cubie model with a little
multiplication table for the twists.

I have always used a facelet
model, except that I number the corners from 1 to 24 and the edges
from 1 to 24 for historical reasons. That is, I started with corners
only or edges only, and have only lately put the two together. It
really does not create any problems for me to use the same numbers
for both edges and corners because the edges and corners
are stored disjointly, as are the edge and corner permutations
for quarter and half turns, and as are the edge and corner
permutations for rotations and reflections.

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.

                        ...  I don't know how a normal 4x4x4 could
be represented though.

I fail to see the problem. Just number the facelets. The only
problem would then lie in deciding what the generators are -- i.e.,
which kind of slice moves do you accept. You would also have to
decide whether to model the invisible 2x2x2 inside, but again if you
did, just number the invisible facelets and include their movements
with your generators.

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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

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