Further Cube Musings
====================

We usually think of positions antipodal to start only, but there are
positions antipodal to any given position.

Given a small enough subgroup of the cube, i.e. one which we can
exhaustively study, it is not hard to determine some examples.

Let's use the square's group and the good ol' pons asinorum.
Pons is antipodal to position X.

Pons + X = Antipode  (let's use position p135)

p135 2 X, 4 T       L2 B2 D2 F2 T2 F2 T2 L2 T2 D2 F2 T2 L2 D2 F2

Solving for position X is easy enough....

X = Antipode - Pons

Position X =  F2 D2 L2 D2 F2 L2 T2 F2 T2 F2 T2 F2 L2

The idea of        (-1) * pons   or  (-pons) is equivalent to
the inverse of pons, since (+pons) + (-pons) = identity.

So Pons and Position X are antipodes of each other. Using this
straightforward method we can find an antipode to any position
in the square's group, or for any other positions in another
small subgroup.

This brings up the idea of a "Rubik's Tour". Such a tour would
touch on a set of interesting patterns within a given subgroup,
or potentially the entire cube group. Of course, "God's Tour"
would not only touch on all the interesting patterns, it would
also sequence all the patterns AND orient them in space such that
the number of q turns would be minimal for the tour! I am currently
working on "God's Tour" for some of the lesser subgroups, touching on
say a dozen patterns for the square's group. If humans and computers
ever resolve "God's Algorithm" there is some solace that there are
problems even more intractible.

Hmmmm, I just had a thought. It would probably be best to group all
the patterns closer to start and work outwards towards the more
antipodal ones.

With the smaller groups a "Total Tour" would be possible! Visit all
elements!

-> Mark <-