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From: "christopher f. chiesa"
Message-Id: <199709031702.NAA20567@cyber1.servtech.com>
Subject: Re: Open and Closed Subgroups of G
To: cube-lovers@ai.mit.edu
Date: Wed, 3 Sep 1997 13:02:11 -0400 (EDT)
Greg Schmidt (SCHMIDTG@iccgcc.cle.ab.com) mentions discomfort about
how concepts of "parity" are applied to the Cube. I second the
notion! :-)
I assume that by "parity" we mean that which is conserved as the "twist"
of corner cubies or the "flip" of edge cubies. I myself have a HELL of a
time determining a particular corner cubie's precise amount (N/3, N an
integer) of "twist," or a particular edge cubie's precise amount (N/2, N an
integer) of "flip," other than in the case of an observable change in ONLY
that particular cubie -- and moreover, ONLY in its ORIENTATION. Any change
in a cubie's POSITION, relative OR absolute, renders my notions of "twist"
and "flip" rather fuzzy.
F'rinstance, start with a Cube in the "solved" state and perform the sequence
(generator?):
R' D2 R F D2 F' U2 F D2 F' R' D2 R U2
You will find that "FRU has been twisted -1/3 ("one 'notch' CCW"), and BLU
has been twisted +1/3 ("one 'notch' CW")," relative to their previous
orientations (i.e., relative to "solved") -- and that this is easy to assess
largely because the "solved" state of the rest of the Cube makes it very
clear how the corner cubies' orientations have changed (and their positions
have NOT). The sequence/generator would produce the same net effect
(twisting FRU -1/3, and BLU +1/3) when performed on the Cube in ANY state; it
would merely be more difficult for the casual observer to identify against
the background of a "scrambled" Cube state.
But, back to the start-from-"solved" example. If I now make the single turn
B'
I no longer find it so easy to characterize the corner-twist parity state of
the Cube, because (all of) the corner-cubies affected by this particular
Cube-state-change have left their previous positions, leaving me to wonder,
"RELATIVE TO WHAT" their twist is to be assessed. How is it done? What can
now be said about the "twist state" of, say, the former BLU (now BRU) cubie?
What about the former BLD (now BLU) cubie?
My efforts to "reason it out," within the limitations of my group-theory
background (which is now infinitely broader thanks to Jerry Bryan!), lead to
what almost seems a paradox. For what it's worth, I present it for your
discussion, and will be very interested to hear what you Cubemeisters are
able to contribute!
Observe that the orientations of all corners in the F layer remain unchanged
by the B' operation last performed. In particular, the FRU cubie retains its
-1/3 twist relative to (what's left of) the "solved" state. Assuming that
the "twist" of a cubie which "hasn't moved" REMAINS THE SAME, as opposed to
being, say, "implicitly redefined" by the movement of OTHER cubies, I can
still say a few things -- though not as many things as I would like! -- about
the twist-states of the corner-cubies in the "B layer" after that B' face
turn.
Invoking twist-parity-conservation (let's just say "twist-conservation,"
okay?), I assert that "the TOTAL twist of all corner cubies in the B layer
must still be 'some integer plus 1/3,'" so as to "cancel out" the -1/3 twist
remaining on FRU. The B' turn thus imparted "some integer" TOTAL twist,
which is to say, a total of 0 "net" twist, to the corner cubies in the B
layer -- but was it e.g. "0, 0, 0, 0" or "+1/3, +1/3, -1/3, -1/3?" (I
believe all other combinations reduce to these.) Note that this boils down
to asking, "does a face turn, if it twists corner-cubies AT ALL, twist ALL
FOUR the SAME WAY (i.e. apply the same "net twist" to all four), or NOT?"
Is there a definitive answer? A standard assumption? Proof or disproof of
either? It seems there would _have_ to be, in order to have "meaningful"
discussions of "twist" at all.
For a while I thought I could prove that it was the "0, 0, 0, 0" case, but it
turned out that one of my working assumptions was equivalent to STATING that
it was the "0, 0, 0, 0" case. I was only "proving" my own ASSUMPTION. Glad
I didn't post THAT. :-)
Naturally, analogous issues and questions will arise when discussing
edge-cubie "flip" and the conservation thereof. :-)
All in all, I'd be VERY interested in seeing the professional theoretical
dissection of this issue!
...
That's all I have today on the subjects of "twist," "flip," and "parity/
conservation thereof." But before I go, I'll leave you with two more
demented, blue-sky thoughts. Beware; this is what I get for reading Star
Trek novels before bed, and again at breakfast...
1) At the edge of my intuition, beyond my ability to formalize, I fancy
I sense that there might be a way of looking at the Cube, perhaps
through the use of additional spatial dimensions or their mathemati-
cal equivalents, in which the Cube is in some sense "always" in the
"solved" state, or at least in which it is trivially obvious where lies
the "direct path" back TO the "solved" state. I'm visualizing some
sort of extra-spatial "rubber bands," or "strings" (in those higher
spatial dimensions specifically so as to avoid "tangling" issues)
that "trace" the route (or "net" route) taken by each cubie, or arbi-
trary collection of cubies, from its/their position(s)-and-orienta-
tion(s) in the "solved" state, to its/their p(s)-and-o(s) in a "scram-
bled" Cube. In such a perception, one could simply "tug on the
strings" and "pull" the Cube back to "solved." Does this make ANY
kind of sense to ANYBODY else here? I feel as though I can "almost
see it."
2) Is there a notion, has anybody done any work, on Cube states which
are each other's "duals?" I define the "dual" of a Cube state X as
that Cube state reached by performing, on a "solved" Cube, the same
sequence of turns/moves which "solve" Cube state X. In other words,
define a sequence of turns which transforms the Cube from state X
to "solved," then apply that sequence again to the "solved" cube to
arrive at state Y. State Y is then the "dual" of state X. Ques-
tions abound:
- does each state have EXACTLY ONE dual? Or many, depending on
the specific sequence (as we know, there are many) of moves
performed in solving state X ? (My gut feeling is that each
state has exactly one dual. This would seem to be pretty easy
to prove using the group-theory math at the disposal of many
readers here.)
- are there states which are their OWN duals? (Yes, clearly;
the trivial "checkerboard" pattern arising from a single 180-
degree turn of each face, is its own dual)
- a state which is its own dual, is a "two-cycle" with the
"solved" state: perform the generating sequence on either and
get to the other. Are there "three-cycles?" "Four-cycles?"
etc.?
Looking forward to the followups,
Chris Chiesa
lvt-cfc@servtech.com