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---------------------------------------- Even more thoughts on "Shift Invariance" ---------------------------------------- >>Mark continues >> >> Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant >> UR11 = U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 U2 R1 U1 R1 U1 R3 U1 R3 U1 R3 >> (22 q or 20 h moves) >> Martin asks: >Is UR11 the shortest process effecting the ``odd'' element in <U,R>?

After a bit of computer cubing I found:

p183 6 Twist R1 U3 R2 U3 R1 D3 U3 R1 U3 R3 D2 R3 U3 R1 D3 U3 (18 q or 16 h moves)

This requires using the larger group of <U1, R1, D1>, although I

expected a 16 turn process. Note the fact this larger group has face

index 3 (rather than 2). But now the process is NOT shift invariant

and we see the route itself can determine whether it will be

shift invariant!

I welcome any mathematical explanation!

With even more contemplation I noticed that the process for

the edge 3-cycle

UR1 = U3 R1 U2 (R1 U1)^2 R2 U3 R3 U3 R2 U1 (16 q, 13 h)

...was reducible to

UR1a= F1 U2 (F1 U1)^2 F2 U3 F3 U3 F2 (14 q, 11 h)]

Of course, now we are using <U1, F1> rather than <U1, R1>.

-> Mark <-

Email: mark.longridge@canrem.com