From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 3 18:01:12 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id SAA17413; Wed, 3 Sep 1997 18:01:11 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From lvt-cfc@servtech.com Wed Sep 3 13:05:50 1997 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