   Date: Wed, 08 Dec 93 14:08:15 -0500 (EST)   From: Jerry Bryan <BRYAN%WVNVM.BITNET@mitvma.mit.edu >
~~~ ~~~ Subject: 1152-fold Symmetry and 24-fold Symmetry

I guess it's time to try to explain what I mean by 1152-fold symmetry
and 24-fold symmetry.

Let me start with two or three very simple ideas. First, consider
two equally colored and oriented cubes at Start. To one of them,
apply F, and to the other one apply R. The obvious solution to the
first one is then F' and the obvious solution to the second on is
then R'. But take both cubes and toss them through the
air to someone else, so that the spatial orientation is lost.
Almost anyone would solve either cube by finding the one face that
was twisted clockwise and simply twisting it counter-clockwise.
No distinction between F and R would be made, and it would be
"obvious" to any reasonable person that the cubes were equivalent.

As a slightly more formal application of this idea, consider again
Start to which R has been applied. We could rotate the whole
cube in space using Singmaster's script-U operation. That is, grasp
the Up (top) of the cube and turn the whole cube in space clockwise.
Now, the cube looks like F has been applied rather than R, and the
solution looks like F' rather than R'. If we applied F', the cube
would be solved, but the colors would be oriented wrong. We could
restore the colors by script-U'. Thus, (script-U F' script U') is
exactly the same thing as R' (we are just using conjugates in a
very simple way).

Continuing in this vein, take any two equally colored and oriented
cubes at Start. To one of them, apply some long sequence of
permutations P. To the second one, apply (script-U P script-U').
At this point, the two cubes are definitely not "equal" in some
sense -- you could clearly tell them apart. Yet, they are
clearly "equivalent" in some sense, because if P' is a solution to
the first cube, then (script-U P' script-U') is a solution to the
second one. Furthermore, it should be obvious that it is not really
necessary to use the (script-U script-U') conjugate on the second
cube. Rather we can think of some rotation as having been performed
on P to give Q, and then of Q as having been performed on Start, so
that the same rotation that was applied to P could be applied to P'
to give Q', and Q' is equivalent to (script-U P' script-U').

If I can wax sophomorically philosophical for a minute, I tend to
think of there being two kinds of permutations in mathematics.
The first is the "permutations and combinations" kind of thing you
run into in probability and statistics. The second is the permutation
operator kind of thing you run into in abstract algebra or group
theory. With this kind of thinking, the cube itself represents the
first kind of permutation, where the cube is an object being operated
on, and the twists of the cube are the second kind of permutation,
where the twists are permutation operators and are doing the operating.
Well, at some deep level, the two kinds of permutations are very much
the same thing, so it is very natural to think of operating on
(rotating, in this particular case) a permutation P, where P itself
is an operator.

I need one more simple idea. Again, think of a cube in Start, and
think of Singmaster's script-U operator. We can (informally) write
script-U = (Front --> Left --> Back --> Right --> Front). But suppose
the cube is colored as Font=Red, Left=White, Back=Orange, Right=Blue).
We could just as well write script-U = (Red --> White --> Orange
--> Blue --> Red). It looks as if for any fixed coloring, we can
freely interchange Singmaster's notation with a notation based on
colors. But we can't really. For example, colored as I described it
above, script-F is equivalent to script-Red. Either is the same as
grasping the front of the cube and rotating the whole cube clockwise.
But first perform script-U. Now, script-F is the same as
script-Blue). The Front/Back/Up/Down/Left/Right notation is fixed in
space, but the color notation is not.

Now, we try to put all this together. Once again, consider two
equally colored and equally oriented cubes in space, and apply F
to the first one and (R script-U) to the second one. Both
cubes can now be described as "Start with the front twisted clockwise
by 90 degrees), but the colors are not the same. They are clearly
equivalent, but under my internal computer model for the cube, they
are not equal. Furthermore, no amount of additional application of
Singmaster's whole cube "script" operators will make them equal.
The only thing that will make them equal will be to rotate the colors.

The exact same thing applies to reflections. Consider two equally
colored and oriented cubes in Start, and apply F to one and F' to
the other. The cubes are equivalent but not equal. Hold up the
cube to which F' has been applied to a mirror. The mirror-image
now has F applied instead of F', but the colors are wrong (they
have been reflected). To make the cubes equal, it is necessary to
reflect the colors of the mirror-image.

Hence, my program generates equivalence classes by applying
a cube rotation, a color rotation, a cube reflection, and a color
reflection. There are 24 cube rotations and 24 color rotations
(one of each is the identity), and any cube rotation can occur with
any color rotation. There are 2 cube reflections and 2 color
reflections (one each is the identity), but the cube reflection
identity must occur with the color reflection identity and vice
versa. Thus, there are (in general) 24x24x2 elements in each
equivalence class. I only store one element of each equivalence
class in my data base. Some of the equivalence classes have fewer
than 24x24x2 elements, namely those for which the cube configuration
inherently has a high degree of symmetry.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax     