A catalog of rainbow moves               6-9-96

                                                             - D Joyner



  Column moves:

  We number the columns as 1,...,8. We will use a signed cycle notation to
denote an action of a move on the columns of the masterball. For example, 
  (a) a move which switches the 1st and 3rd column but flips both of them
over will be denoted by (1,3)_,
  (b) a move which sends the 4th column to the 6th column, the 6th column 
to the 5th column, and switches the 2nd and 3rd column but flips both of 
them over will be denoted by (2,3)_(6,5,4),


              move                 cycle 


               f1                 (1,4)_(2,3)_  
               f2                 (2,5)_(3,4)_  
               f3                 (3,6)_(4,5)_  
               f4                 (4,7)_(5,6)_  
               f5                 (5,8)_(6,7)_  
               f6                 (1,6)_(7,8)_  
               f7                 (2,7)_(1,8)_  
               f8                 (3,8)_(1,2)_  
            f1*f2*f1              (1,2)_(3,5)
            f1*f2*f1*f2           (5,4,3,2,1)_
            f1*f3*f1              (1,5)(2,6)
            f2*f3*f2              (2,3)_(6,5,4)
            f1*f4*f1              (1,7)(5,6)
            f1*f5*f1              (5,8)_(6,7)_
            f1*f8*f1              (2,8)(3,4)_
            f8*f1*f8              (1,8)_(4,3,2)
            f2*f1*f2              (1,3)(4,5)_
            f3*f1*f3              (1,5)(2,6)
            f3*f6*f3              (1,3)(4,8)_
            f8*f1*f2              (1,4)_(2,3,8,5)_



  
  Some products of 2-cycles on the facets:

   These are all based on an idea of Andrew Southern. The polar2swap and
equator2swap were obtained by trying variations of some of Andrew's moves 
on a MAPLE implementation of the masterball.
   We number the facets in the i-th column, north-to-south, as i1, i2, 
i3, i4 (where i=1,2,...,8).


                     move                          cycle 

          r1*f4*r1^(-1)*r4*f4*r4^(-1)          (44,84)(41,81)
         f1*r1*f4*r1^(-1)*r4*f4*r4^(-1)*f1     (14,84)(11,81)
                    polar2swap36                 (11,14)(31,61)
                   equator2swap36                (12,13)(32,62)
                   equator2swap18                (62,63)(12,82)


 where 
 
                            polar2swap36=
              f1*r3^(-1)*r4^(-1)*f1*f2*r1*r4^(-1)*f2*r4^4*f2*
                  r1^(-1)*r4*f2*r4^4*f1*r3*r4*f1

 (if you replace r3 by r2 both times in this move you get the same
 effect),
                            polar2swap18=
              f1*r3^(-1)*r4^(-1)*f3*f4*r1*r4^(-1)*f4*r4^4*f4*
                  r1^(-1)*r4*f4*r4^4*f3*r3*r4*f1
 

                           equator2swap36=   
  f1*r4^(-1)*r3^(-1)*f1*f2*r2*r3^(-1)*f2*r3^4*f2*
                  r2^(-1)*r3*f2*r3^4*f1*r4*r3*f1


                           equator2swap18=
   f1*r4^(-1)*r3^(-1)*f3*f4*r2*r3^(-1)*f4*r3^4*f4*
                   r2^(-1)*r3*f4*r3^4*f3*r4*r3*f1