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~~~ Subject:

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 - PonsPosition 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 <-