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Date: Tue, 14 Dec 1993 20:50:51 EST
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Re: Symmetry
In-Reply-To: Message of 12/13/93 at 22:31:31 from hoey@aic.nrl.navy.mil
On 12/13/93 at 22:31:31 hoey@aic.nrl.navy.mil said:
>In the absence of face centers, there is another kind of reduction
>that takes account of the 24 possible positions of the resulting
>collection of edges in space. So two positions X and Y are considered
>equivalent if
> X = m' Y m c
>where m is a rotation or reflection in M, and c is a rotation.
>My understanding of Jerry Bryan's method is that he combines "m c"
>into a single rotation or reflection, and factors out the reflection
>on both sides. It seems to me that what he calls a "color rotation"
>is premultiplication, while a "cube rotation" is postmultiplication.
>[I am somewhat uncertain of this, because it doesn't explain how there
>can be a 1252-element symmetry group when face centers are present, so
^^^^
should be 1152
>perhaps I'm missing something crucial.]
I just reread "Symmetry and Local Maxima". Let's see if I can make
some sense of this. I believe "pre-multiplication" and
"post-multiplication" are correct. In my computer model, the
corner facelets are simply numbered from 1 to 24, and any configuration
of the corners is an order-24 row vector. The rotation and reflection
operators are also order-24 row vectors, again with each cell simply
containing a number from 1 to 24.
In almost anybody's programming language you would copy an order-24
row vector with something like
For i = 1 to 24 B(i) = A(i)
Well, if P is a rotation operator, you could perform a rotation
two ways. I guess one is pre-multiplication and one is
post-multiplication.
1) For i = 1 to 24 B(i) = A(P(i))
2) For i = 1 to 24 B(i) = P(A(i))
(As an aside, this illustrates the question I raised in my previous
post about "which is the operator and which is the thing being
operated on?" Is P operating on A, or is A operating on P?)
In fact, if what I am doing is properly called pre- and post-
multiplication, then I am doing both as a part of a single,
composite operator. I.e.,
For i = 1 to 24 B(i) = P(A(P(i)))
More completely, there are 24 rotations, P1 through P24, so the
actual loop looks something like
For j = 1 to 24 for k = 1 to 24
for i = 1 to 24 Bj,k(i) = Pj(A(Pk(i)))
Finally, if Q is a reflection (actually, if Q1 is the identity and
Q2 is the reflection), then we have
For j = 1 to 24 for k = 1 to 24 for m = 1 to 2
for i = 1 to 24 Bj,k,m(i) = Qm(Pj(A(Qm(Pk(i)))))
I believe this loop calculates Dan Hoey's M. In my data base, I
store the minimum of Bj,k,m over j = 1 to 24, k = 1 to 24, and
m = 1 to 2. I tend to call the minimum of Bj,k,m a canonical
form. I am not sure if that is the best terminology. The
minimal element is not any simpler than any other. It is just
that I need a function to choose an element from a set, and
picking the minimal element seems very natural. Any other
element would do as well, provided I could always be sure of
picking the same element.
Also, my criterion for equivalence is slightly
different (but isomorphic, I think) than the one described by
Dan Hoey. Suppose A and B are two cubes.
Rather than mapping A to B or B to A in M, I map both A and B
to their respective canonical forms. A and B are equivalent if
their respective canonical forms are equal.
I hasten to add that the actual loop in the program is a bit
more complex than the one shown above. The one above would
be far too slow. The actual loop is several hundred times
faster.
Now, as to the centers. I still sometimes have a certain doubt
about the centers. They are fixed, so how can you reduce the
problem (i.e., increase the size of the equivalence classes)
by both rotating the cube and rotating the colors (by both pre-
and post-multiplication)?
In my computer model for the centers, I simply number center facelets
from 1 to 6, and the centers are stored as an order-6 row vector.
The centers are disjoint from the corners (as well as from the edges),
so there is no problem in numbering one set of objects from 1 to 24 and
another from 1 to 6.
I define a set of 24 rotation operators P* on the centers, corresponding
to the 24 rotation operators P on the corners, and a set of 2 reflection
operators Q* on the centers, corresponding to the 2 reflection operators
Q on the corners. Then, if C is an order-6 row vector representing
the centers, I calculate Dj,k,m = Q*m(P*j(C(Q*m(P*k)))) anytime
I calculate Bj,k,m = Qm(Pj(A(Qm(Pk)))).
(Read the asterisks above as superscripts. I am not intending
the multiplication operation which the asterisk denotes in
many programming languages.)
Hence, I rotate and reflect the centers right along with the corners.
But there are only 24 distinct states for the centers, and each can
occur with any canonical form for the corners. Hence, the "corners
plus centers" data base is exactly 24 times larger than
the "2x2x2" data base.
My model for the cube seems to start out 24 times larger than
everybody else's. However, by storing only the canonical form
for each equivalence class, and since most of the equivalence
classes have 1152 elements, my data base seems to end up about
48 times smaller than everybody else's. This fact seems to
remain true, even when the "fixed centers" are added in.
I am not sure if this answers Dan's question about my model
with centers added. Effectively, I am using a "fixed corners"
representation of the cube, and rotating the centers. Each
equivalence class for the corners under M has (up to) 1152
elements, and each equivalence class for the centers under
M has only 24 elements. But it doesn't seem to matter.
(Up to) 48 different configurations of the corners within
M share each configuration of the centers.
Since I am in this deep, let me finish explaining certain details
of my model. I don't really store all 24 elements of each
row vector. I really just store 8. That is, I store the
facelets for the front and back face. The other 16 facelets
can be reconstructed from the first 8. In effect, storing
a number from 1 to 24 stores both the location of each cubie
and its twist. Finally, I really, really only store 7 elements.
In the canonical form, the first element is always 1, so there
is no reason to store it. Thus, a data base record for the
2x2x2 looks like
CCCCCCC,L
where the CCCCCCC are the seven elements representing the canonical
form, and L is the corresponding level.
When you add the centers, I started out with notion that the
order-6 row vector for center only has 24 possible states. Thus,
it can be encoded as a number from 1 to 24. This lead to the
following
CCCCCCC,L,R
where CCCCCCC and L are as before, and R is an index encoding
the orientation of the centers. But this can be improved upon
even further. With my model for the corners plus centers, each
distinct value of CCCCCCC will occur exactly 24 times, and each
distinct value of CCCCCCC is already represented in my data
base for the 2x2x2. Hence, I can have the exact same number of
(longer) records, and encode the corners of the 3x3x3 as
CCCCCCC,L,L1 L2 L3 .... L23 L24
where CCCCCCC is as before, L is the level of CCCCCCC in the
2x2x2, and L1 through L24 are the levels of CCCCCCC in the corners
of the 3x3x3 when the index of the position of the centers with
respect to the corners is 1 through 24, respectively. Hence, my
data base for the corners of the 3x3x3 has the same number of
records as the data base for the 2x2x2, and is physically only
four times larger.
>With regard to the edge cube, I should mention that no one has
>mentioned a 9 QT process for the all-flip nor a 15 QT process for the
>pons-asinorum-all-flip. Of course, the latter would be somewhat more
>interesting, being the longest optimal sequence.
I will work on these two cases, but it will take some time. My model
is very good at storing a great many states of the cube very
compactly, but it does not encode processes at all. I will have
to extract the processes by hand. This is quite easy in my
data bases for the 2x2x2 and corners of 3x3x3. But it is quite
hard for the edges because the data base is 4.2 gigabytes.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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?