From cube-lovers-errors@mc.lcs.mit.edu Thu Sep 4 17:02:06 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id RAA24202; Thu, 4 Sep 1997 17:01:53 -0400 (EDT) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From jbryan@pstcc.cc.tn.us Thu Sep 4 12:54:09 1997 Date: Thu, 04 Sep 1997 12:50:11 -0400 (Eastern Daylight Time) From: Jerry Bryan Subject: Re: Open and Closed Subgroups of G In-Reply-To: <199709031702.NAA20567@cyber1.servtech.com> To: "christopher f. chiesa" Cc: cube-lovers@ai.mit.edu Message-Id: On Wed, 3 Sep 1997, christopher f. chiesa wrote: > 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: The concept of "dual" which you are describing is standard in group theory (and be extension, in cube theory). A "dual" is properly called an inverse. If you have a sequence of turns which creates a position, the inverse sequence consists of writing the turns in reverse order, and converting clockwise turns to counterclockwise turns and vice versa. So the inverse of FRU' is UR'F'. If there are multiple sequences for a position (and most typically there are), you can do the same thing for any such sequence. Also, a position can be described in terms of which cubies have gone where. For example, you might have something like flu --> fur fur --> frd frd --> fdl fdl --> flu (flu is the front-left-up cubie etc. Standard Singmaster notation uses lower case letters for cubies and upper case letters for the moves themselves.) You could get the inverse by reversing the arrows like so. flu <-- fur fur <-- frd frd <-- fdl fdl <-- flu More commonly, you would write the inverse by swapping the cubie designations between the left and right side of the arrows like so. fur --> flu frd --> fur fdl --> frd flu --> fdl I don't know what you mean by "any work", but here are some standard information about inverses. The length of a position X is the same as the length of its inverse X', where length is the minimum number of moves to create the position. If X' is the inverse of X, then X is the inverse of X'. The symmetry of an inverse X' is the same as the symmetry of a position X (see Symmetry and Local Maxima in the archives for a discussion of symmetry). A local maximum is a position such that no matter which move you make, you will be one move closer to Start. It is not necessarily the case that the inverse of a local maximum is also a local maximum. > > - 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 ? Yes, inverses are unique, both for groups in general, and for cubes in particular. > > - 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) You have answered your own question. Many positions are their own inverse. Some of them are much more complicated than the one which you describe. > > - 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.? > The proper term for the concept you are describing is order. If you repeat a maneuver n times from Start and return to Start, then the position is of order n. (Strictly speaking, the order of a position is the smallest n which will work. Obviously, if n will work then so too will 2n, 3n, etc.) There are many different orders for which there are cube positions of that order. One of David Singmaster's early Cubic Circulars (I don't have the reference handy) had a table of possible cube orders and how many positions there were of each order. The term cycle is also very important in group theory (and by extension in cube theory). Suppose you look at a scrambled cube and determine that cubie a has gone to cubie b's place, cubie b has gone to cubie c's place, and cubie c has gone to cubie a's place, then a, b, and c form a 3-cycle. The way I have defined this particular 3-cycle, you could write it as (a,b,c), as (b,c,a), or as (c,a,b). This so-called cycle notation is circular, so it does't really matter which you write first. However, (a,c,b) is a different cycle than (a,b,c). In fact, (a,c,b) is the inverse of (a,b,c). Just for emphasis, (a,b,c) is not like an ordered pair (or really an ordered triple in this case). (a,b,c) means a goes to b, b goes to c, c goes to a. As an example of a cycle in purely cube terms, the cycle for the example I gave earlier would be (flu,fur,frd,fdl), so it is a 4-cycle. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7198 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990