Notes on the UR Group
---------------------

Well, some small news about the < U, R > group. Previously I believed
that my 3-cycle of edges:

UR1 = U3 R3 U2 R1 U1 R3 U1 R1 U1 R1 U2 R3 U3 R1 U3 R3 = 18 q, 16 h

...discovered by hand was minimal. My newly created < U, R > solver
(now being at the point of churning out correct results) as
happily proven me wrong!

UR2 = U3 R1 U2 R1 U1 R1 U1 R2 U3 R3 U3 R2 U1 = 16 q, 13 h

Also I found 6 "UR-Reflective" processes altogether. This is all
there are up to and including 12 q turns:

U3 R1 U1 R1 (U2) R3 U3 R3 U1 = R3 U1 R1 U1 (R2) U3 R3 U3 R1   (10)
U1 R3 U3 R3 (U2) R1 U1 R1 U3 = R1 U3 R3 U3 (R2) U1 R1 U1 R3   (10)
               ( U2 R2 ) ^ 3 = ( R2 U2 ) ^ 3                  (12)
  U1 R1 U2 R3 U2 R3 U2 R1 U1 = R1 U1 R2 U3 R2 U3 R2 U1 R1     (12)
          ( U1 R1 U3 R3 ) ^3 = ( R1 U1 R3 U3 ) ^3             (12)
          ( U3 R3 U1 R1 ) ^3 = ( R3 U3 R1 U1 ) ^3             (12)

The program is still a sluggish beast, but I think with further
refinements it should eventually find other interesting results
like antipodes and pure twists, etc.

-> Mark <-

Email: mark.longridge@canrem.com