Date: 2 August 1980 01:55-EDT
From: Alan Bawden <ALAN at MIT-MC>
Subject:  a metric for the cube group.
To: CUBE-HACKERS at MIT-MC, McKeeman at PARC-MAXC

First off, a metric is a function (call it D) that assigns a
non-negitive number to every pair of points in some set.  This number
is to be thought off as the distance between those two points.  The
function must satisfy the following:

For all a, b and c
1) D(a,b) >= 0
2) D(a,b) = D(b,a)
3) D(a,b) = 0 if and only if a = b
4) D(a,b) + D(b,c) >= D(a,c)

(Number 4 is usually called the "triangle inequality".  It is the
constraint that most makes D act like a distance, and not something
random.)

We wish to construct a metric on the set of all attainable cube
configurations.  So from now on, those lower case letters will
represent cube configurations.

Now we have recently been talking a lot about methods of counting the
number of "twists" that it takes to perform certain manipulations of
the cube.  We have been looking for a function (call it T) that
assigns a non-negitive integer to each manipulation.  I claim that it
is obvious that any such function should satisfy the following:

For all M and N
1) T(M) >= 0
3) T(M) = 0 if and only if M = I	(I is the identity manipulation)
4) T(M) + T(N) >= T(MN)

(We adopt the convention of using upper case letters to represent
manipulations.  Also we shall use M' to denote the inverse manipulation
from M.)

Now manipulations can be applied to configurations to yeild other
configurations.  We use aM to denote the configuration resulting from
applyint the manipulation M to the configuration a.  (Note that
(aM)N = a(MN), so we may omit the parens and simply write aMN.)

Now how may we use our twist measuring function T to obtain a metric
on the configurations?  Again I think it is obvious that we wish the
relationship D(a,aN) = T(N) to be true for all configurations a, and
all manipulations N.

It is easy to show that given that D(a,aN) = T(N), metric property
number 1 is equivalent to twist measure property number 1.  Similarly
for numbers 3 and 4.  But what about metric property number 2?

Well, if T(N) = D(a,aN), and D(a,aN) = D(aN,a) (property 2!), and
a = aNN', then we have that T(N) = D(aN,aNN') = T(N').  So the missing
property of twist measures must be that T(N) = T(N').

So this means that if we agree that T(L) = 1, and we like metrics
(how can we use words like "distance" unless we have a metric?), then
T(LLL) = T(L') = T(L) = 1.  We can argue about T(LL) some other time!