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To: cube-lovers@ai.mit.edu
Subject: Some notes on the antislice group.
Date: Sat, 16 Dec 95 01:22:05 +0100
From: Michiel Boland
This is something for the holidays.
I hope someone finds it interesting :)
Some notes on the antislice group.
Notations: Ua = UD, U2a = U2D2, U'a = U'D', Ra = RL, Fa = FB, etc
Consider the three `slices' of edge cubicles (one slice
containing UR, RD, DL, LU, another containing UF, FD, DB and BU,
and another containing FR, RB, BL, LB).
Any operation in the antislice group (the subgroup of G
generated by Ua, Ra and Fa) will map each slice to another
slice. Also, if we restrict ourselves to antislice movements, we
can define an orientation for each slice (choose a fixed cubie
in each slice and define the orientation of the slice to be the
orientation of that cubie).
A fairly obvious subgroup of the antislice group is the one in
which all three slices are in their original position *and* are
oriented correctly. I have been giving this subgroup, which is
in fact a normal subgroup of the antislice group, some study.
To make speaking a little bit easier, I will use the letter T
for this group (don't ask me why :)
If one takes a cube in position START, and applies an operation
in T to it, one finds that, if one looks at a face, each facelet
has either the colour of that face's center, or the colour of
the opposite face's center. Therefore, the patterns generated by
movements in T can be deemed `pretty'.
Some patterns that can be generated from transformations in T
are the Pons Asinorum (U2a R2a F2a), 4 Plusses (Ua Ra U2a Ra Ua F2a),
and 6xH (Ua Ra U2a F2a Ra U'a).
A pattern that cannot be generated from T-movements is the
four-dot pattern (I'm just stating this as a fact; I still
haven't got a proof for it.)
The group T contains 256 elements. It is isomorphic to C_2^8
(the cartesian product of eight copies of C_2). Hence, each
element of T is its own inverse.
The group T is generated by the following elements:
U2a
R2a
F2a
Fa U2a F'a
Ua R2a U'a
Ra F2a R'a
Ua Ra Ua Ra Ua Ra
Ra Fa Ra Fa Ra Fa
Note that Fa Ua Fa Ua Fa Ua = Ua Ra Ua Ra Ua Ra Ra Fa Ra Fa Ra Fa
There are two obvious metrics on the antislice group (and on T):
the `quarter' turn and the `half' turn metric.
It takes at most four `quarter' anti-slice turns to get from any
position in the antislice group to a pattern in T (Ua Ra Fa Ua
is a maximal case in this respect.)
If one groups the members of T by their lengths in either metric
one gets some interesting results.
length in quarter-turn metric
0 2 4 6 8
+---------------------------+
0 | 1 | 1
1 | 3 | 3
length in 2 | 3 | 3
half-turn 3 | 12 1 | 13
metric 4 | 18 | 18
5 | 15 | 15
6 | 192 11 | 203
+---------------------------+
1 3 15 226 11 256
The eight elements on the `diagonal' form a subgroup of T
(generated by U2a, R2a and F2a).
If one excludes the 192 elements from row 6 column 6, one also
gets a subgroup of T with 64 elements (generated by the three
elements mentioned above, and FaU2aF'a, UaR2aU'a, and RaF2aR'a).
Each of the 192 elements in the 6th row, 6th column can be
uniquely written in the form
X Ya Xb Yc Xd Ye
where X and Y are either Ua, Ra or Fa, X and Y are different,
and a,b,c,d,e are either 1 or -1 (these are meant to be
exponents). (examples: Ua Ra Ua Ra Ua Ra, Ua R'a Ua Ra U'a Ra)
The 11 elements in row 6 column 8 are:
Ua Ra U2a F2a Ra U'a (6xH)
Ua Fa U2a R2a Fa U'a ( ' )
Ua Ra U2a Ra Ua F2a (4x+)
Ra Fa R2a Fa Ra U2a ( ' )
Fa Ua F2a Ua Fa R2a ( ' )
Ua Ra U2a F2a Ra Ua (2xH, 2xDot, 2x+)
Ua Fa U2a R2a Fa Ua ( ' ' ' )
Ra Fa R2a U2a Fa Ra ( ' ' ' )
Ra Ua R2a F2a Ua Ra ( ' ' ' )
Fa Ua F2a R2a Ua Fa ( ' ' ' )
Fa Ra F2a U2a Ra Fa ( ' ' ' )
--
Michiel Boland
University of Nijmegen
The Netherlands