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