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Date: Fri, 17 Dec 1993 00:54:00 EST
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Some Additional Distances in the Edge Group
It is now known that using the qturn metric, Start has a
unique antipode in the edge group, namely Mirror-Image-
of-Edges-Flipped. The antipode is 15 qturns from Start.
Also, I have a complete data base of equivalence classes
in the edge group documenting the distance from Start for
each configuration of the edges.
It seems to me that given these two facts, some additional
distances can be determined. For example, it is possible
to determine the distance from any configuration to
Mirror-Image-of-Edges-Flipped. Let Z be a sequence of operators
that converts Start to Mirror-Image-of-Edges-Flipped, and let
A be any configuration of the edges. Then apply Z' to A, look
up the result in the data base of distances from Start, and
that will be the distance from A to Mirror-Image-of-Edges-Flipped.
The reason is quite simple. Let P be a sequence which takes
Z'(A) to Start. Then, Z'PZ takes A to Mirror-Image-of-Edges-Flipped.
This is a very nice use of conjugates.
Another consequence of this result is the following: suppose you
began with Mirror-Image-of-Edges-Flipped and performed a
breadth-first exhaustive search. Start would be antipodal, and
the number of nodes at each level of the tree would be identical
to the existing tree which begins at Start.
In addition, all of the above applies to Mirror-Image-of-Start
and Edges-Flipped with respect to each other. They are
mutually antipodal, and are 15 qturns apart. A tree built with
either at the root would have exactly the same number of nodes
at each level as the existing tree with Start at the root.
Finally, the distance of any configuration from Mirror-Image-of-Start
or Edges-Flipped can be determined. Let Y be a sequence of operators
which converts Start to Mirror-Image-of-Start, and let X be a sequence
of operators that converts Start to Edges-Flipped. Let A be any
cube. Then, the distance of A from Mirror-Image-of-Start
is the same as the distance of Y'(A) from Start, and
the distance of A from Edges-Flipped is the same
as the distance of X'(A) from Start.
I have the sensation in describing this that the Edge group is
square, with Start and Mirror-Image-of-Edges-Flipped 180 degrees
apart, and Mirror-Image-of-Start and Edges-Flipped at the other
two corners of the square.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) (304) 293-5192
Associate Director, WVNET (304) 293-5540 fax
837 Chestnut Ridge Road BRYAN@WVNVM
Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU
If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?