From:

~~~ ~~~ Subject:

Our normal definition of symmetry for Rubik's cube is based

ultimately on the 48 symmetries of the standard math book

wire model of a cube, and the 48 symmetries were discovered

long before Rubik's cube was ever dreamed of. This note is

based on the conviction that these 48 symmetries do not really

capture all that we might think of as "symmetry" when we think

of Rubik's cube. This note has the further purpose to propose

a more general definition of symmetry for Rubik's cube.

I want to start with a couple of really basic concepts. I think

every reader of this list knows what a permutation is, namely

it is a one-to-one onto function on a set. In the case of a finite

set as we have with Rubik, a function on a set is one-to-one

if and only if it onto, so we can sometimes get by with speaking

only of one-to-one or by speaking only of onto.

But what is a symmetry? A very standard definition is something

like "the set of all rigid motions that transform a given geometric

figure onto itself" (James and James Mathematics Dictionary).

Another way to say it is that the transformation preserves the

figure.

Working with that definition, a symmetry almost inevitably may

be interpreted as a permutation. With simple geometric figures,

the permutation would usually be described as being a permutation on

Euclidean n-space -- 2-space for a square or circle, etc., and

3-space for a cube or sphere, etc. Hence, we might think of

a symmetry as being a special kind of permutation, namely one that

preserves a geometric figure in Euclidean n-space.

I have had a difficult time finding books that address the issue

of symmetry vs. permutation to my satisfaction. It is very hard

to think of a symmetry abstractly enough that it doesn't simply

turn into a permutation right before your eyes.

Paul Yale's "Geometry and Symmetry" doesn't really seem to define

a symmetry (it sort of assumes you know what one is), but it does

describe the relationship between a symmetry and a permutation.

I would paraphrase as follows. Label your geometric figure in

some fashion -- e.g., label the edges, label the axes, label

the vertices, or label *something*. Then, there is a homomorphism

between the set of symmetries and the corresponding set of

permutations on the labels.

But I repeat that it is hard for me to conceive of the set of symmetries

in a sufficiently abstract fashion that the symmetries themselves

aren't already permutations on *some* set or other. So it seems to me

that Yale could just as well be talking about homomorphisms between

one set of permutations and another set of permutations as talking

about homomorphisms between symmetries and permutations.

A couple of quick additional points, and then I will go on:

1) since we are talking about homomorphisms, it is obvious that

both the set of symmetries and the set of permutations to which

they map are groups, and 2) most homomorphisms between symmetries

and permutations turn out in fact to be isomorphisms. This latter

observation gives added weight to the notion that symmetries are

just a special kind of permutation.

Given all that has been said so far, we could informally say that

the normal definition of a symmetry is that it is a permutation

that preserves a geometric figure. Our more general definition

will simply be that a symmetry is a permutation that preserves

some property. If we were sufficiently liberal in our notion of

"preserving some property", then most any permutation could be

interpreted as a symmetry. We will not be quite that liberal

by the time we are done, but we will be more liberal than

would be permitted by the standard 48 math book symmetries of

the cube.

But what property of Rubik's cube should we try to preserve if we

want a more general definition of symmetry than the normal one?

I wish to motivate our definition of that property in several

steps.

The standard Rubik's cube definition of symmetry for a position X is

Symm(X) is the set of all m in M such that X=m'Xm, or equivalently

such that mX=Xm. M is the set of 48 permutations on the Rubik's

cube corresponding to the 48 symmetries of a cube.

Write a position Z as Z=XY, where X is the permutation on the

corners and Y is the permutation on the edges. We have

Symm(Z)=Symm(XY)=Symm(X) intersect Symm(Y). For example, we

could have Symm(X)=M, Symm(Y)=I, and Symm(Z)=Symm(XY)=I.

Such a position would look very "symmetrical" because the

corners would be fixed (or "solved"), although the edges

would be scrambled. Most people would consider such a position

to be more "symmetrical" than one where both the corners and

edges were scrambled, although we would have Symm(Z)=I

in either case.

A couple of points before proceeding: 1) From an information

theory point of view, Symm(X) and Symm(Y) separately contain

more information than does Symm(XY). There is an obvious

loss of information when we calculate Symm(X) intersect Symm(Y).

This is a strong indication that Symm(XY) does not tell us

everything we might like to know about the symmetry of a position.

2) The set of positions Z=XY for which Symm(X)=M forms a group

(as does the set of positions for which Symm(Y)=M). This anticipates

where we are headed, namely that group membership is the property

that we should seek to preserve in a more general definition of

symmetry.

A third (and equivalent) definition for Symm(X) is that Symm(X) is

the set of all m in M such that X'm'Xm=I. Most readers will

recognize X'm'Xm as a commutator. Per Dan Hoey, we can generalize

and define CSymm(X) to be the set of all m in M such that X'm'Xm

is in C, the set of 24 rotations of the cube. For example, if

we have Z=XY as before, then CSymm(X)=M means that the corners

are positioned properly with respect to each other, although they

might be rotated with respect to the fixed face centers.

Such a position would look fairly "symmetrical", even to a

non-cubemeister, even though we might have Symm(Z)=I.

Again, we have the set of all positions for which CSymm(X)=M

forms a group. Similarly, the set of all positions for which

CSymm(Y)=M forms a group, and the set of all positions

for which CSymm(Z)=CSymm(XY)=M forms a group.

Recall that there are 98 subgroups of M. For each subgroup K of

M, there is a corresponding subgroup of G consisting of all the

K-symmetric positions. So would could just as well define

symmetry in terms of these 98 subgroups of G. But there are far

more than 98 subgroups of G. (We don't know how many, and I

doubt than even GAP could tell us). Why not simply define

symmetry in G in terms of subgroup membership in G? The symmetry

of a position X is then the set of all subgroups of H of G

for which X is in H. And a symmetry operation (in the sense

that a symmetry is a permutation that preserves something) is

an operation that preserves subgroup membership.

That pretty much completes my proposal, but I have a few closing

remarks.

1) The proposed general definition of symmetry is

analogous to the Thistlethwaite algorithm for solving

the cube. Typical cube solutions gradually solve more

and more of the cube. The "more and more of the cube"

that gets solved can be characterized as a sequence of

nested subgroups. Thistlethwaite reversed the process

and created a sequence of nested subgroups

which in turn solves more and more of the cube. Similarly,

the standard definition of symmetry implies a set of 98

subgroups of G. We reverse the process and let all the

subgroups of G define symmetry instead.2) The proposed general definition of symmetry has the

virtue that it includes the standard definition as a

special case, since the 98 K-symmetric subgroups of

G are in fact subgroups of G.3) The proposed general definition of symmetry has the

virtue that there is only one position that is

"completely symmetric", namely Start itself (the

identity permutation). The standard definition of

symmetry has four positions which are "completely

symmetric", which to me is an unsatisfactory state

of affairs.

(Recall that we have Symm(X)=M for Start, Pons Asinorum,

Superflip, and the composition of Pons and Superflip.

I am still bummed out that this is the case while at the

same time only Start and Superflip are in the center of

G. This suggests that Superflip is "more symmetric"

than Pons. I wonder if such a suggestion would be

supported by my proposed general definition 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