   Date: Sat, 09 Sep 95 09:51:56 -0400   From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: A Proposal for a More General Definition of Symmetry

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
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?)

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