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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 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU

If you don't have time to do it right today, what makes you think you are

going to have time to do it over again tomorrow?