A topspin commutator : RR F LL F
The "commutator" of R^2 and F is
do right twice RR
do flip F
undo RR, i.e. LL
undo F, i.e. F
so the sequence we're talking about is six moves, RRFLLF .
Using the virutal TopSpin at
http://cs.marlboro.edu/courses/fall2011/rubik/topspin
and starting with 1 through 4 inside the flipper.
Typing the flipper as [ ... ] we have :
start : ... 18 19 20 [ 1 2 3 4 ] 5 6 7 ...
RR ... 16 17 18 [ 19 20 1 2 ] 3 4 5 ...
F ... 16 17 18 [ 2 1 20 19 ] 3 4 5 ...
LL ... 18 2 1 [ 20 19 3 4 ] 5 6 7 ...
F ... 18 2 1 [ 4 3 19 20 ] 5 6 7 ...
Looking at this more closely :
* 18, 17, 16, ... are not changed.
* 5, 6, 7, ... are not changed.
* The remaining 6 numbers move :
19 20 1 2 3 4 => 2 1 4 3 19 20
If you do now try this sequence of moves (RR F LL F)
over and over, you'll find that everything
returns with 3 repeats.
Why?
Tracking the paths where the digits go individually
gives this. (This can be confusing - discuss.)
1 => 20's original place
20 => 4's original place
4 => 1's original place
which is a full cycle that returns to its beginning
1 => 20 => 4 => 1
so that in 3 moves, the 1 returns to it's spot.
Similarly,
2 => 19's place,
19 => 3's place,
3 => 2's place
or
2 => 19 => 3 => 2
We'll see other ways to notate these things
(which are called "permutations") real soon.
Like today.
