This is a good question!
We'll start with a comparatively simple answer, and then go from there. What follows here first is a purely mechanical procedure for computing the time-reversal of a pattern. We will use Rubenstein's Revenge as our example.
Rubenstein's Revenge: 5iB 2iU 2oA 3iU 3oAFirst, write it all in reverse order ...
Reverse order ... : 3oA 3iU 2oA 2iU 5iBNext, change all "i"s to "o"s and vice versa and change all "A"s to "B"s and vice versa
From ... 3oA 3iU 2oA 2iU 5iB
to ... 3iB 3oU 2iB 2oU 5oA
Now move all "A"s, "B"s and "U"s one place earlier, moving
the first one, which would sort of fall off the front, to
the last place ...
From ... 3iB 3oU 2iB 2oU 5oA
to ... 3iU 3oB 2iU 2oA 5oB
So far this is exactly the same as our previous reversing
procedure, except that it's being done in the new notation
rather than on the diagram. The last few steps take into
account the SiteSwap we have.
Move each number two places right, moving the ones that fall off the end back on at the front ...
From ... 3iU 3oB 2iU 2oA 5oB
to ... 2iU 5oB 3iU 3oA 2oB
And for now the last step. Take each number and move it
backwards. The amount by which to move it is given by the
number itself. For example, move a 2 backwards by two
places, and move a 3 backwards three places. When you fall
off the front, come back on at the end. This will mean that
the 5 will stay where it is.
From ... 2iU 5oB 3iU 3oA 2oB
to ... 3iU 5oB 2iU 2oA 3oB
That's it !!
You may wonder if and why this last bit always works. After all, since the numbers move by different amounts, why are we never unlucky and end up with two numbers being moved to the same place? Well, it turns out that if you do end up with that sort of clash, then the original sequence of numbers can't have been a valid SiteSwap. This is _precisely_ the requirement for a sequence to be a valid SiteSwap, so if you start with a valid SiteSwap, it never goes wrong.
Well, of course, we've given little or no justification for most of this, and you might wonder where it all came from. As we said above, the origins of the first few steps can be traced back to the time-reversal on the diagram. The last two steps are derived from the time-reversal of an ordinary SiteSwap. To see the comparison, here is that process applied to the SiteSwap "1 2 3 4 5" ...
First, write it backwards ...
5 4 3 2 1
Now, from each place count backwards by the amount given by
the SiteSwap value. For example, from the 2, count backwards
by two.
From ... 5 4 3 2 1
to ... 5 2 4 1 3
That's it. You can see the similarity in the processes. The
main difference is that in the full process above we shift
the numbers by two places. A shift like that is irrelevant
in ordinary SiteSwaps, but here it serves to make sure that
the SiteSwap values are lined up correctly with the arm
movements.
We can see the whole process in action on the full diagram.
+----+ ###### +----+
| Lr | # Lr # 5+> | Ll | Here we've drawn the diagram
+----+ ###### +----+ for Rubenstein's revenge,
^ 3 ^ 2 although it has been a some-
o + o + what compressed. We've also
2 v 3 v doubled up on those states
+----+ +----+ +----+ +----+ that occur twice in the
| Ul | | Ul | | Ur | | Ur | pattern, for clarity.
+----+ +----+ +----+ +----+
^ 3 ^ 2 When we work out the time-
+ o + o reverse we exchange every
2 v 3 v "r" for "l" and vice versa,
+----+ ###### +----+ we reverse the arrows, and
| Rr | <+5 # Rl # | Rl | we exchange "o"s and "+"s.
+----+ ###### +----+
So here's the time-reversal,
+----+ +----+ +----+ but where should we put the
| Ll | | Ll | | Rr | | Rr | been that ball that had been
+----+ +----+ +----+ caught on the previous left
hand exchange, shown at left.
So, the "5" thrown in the normal-time pattern from the state
"Lr" becomes a "5" thrown from the marked state above in the
time-reversal.
So that's how we compute the time-reversal of a SiteSwap with hand movements, and it brings us close to the end of the tutorial series. In our next lesson we'll review some of what we've seen, and take a second look at some of the details and terminology.
Go to Lesson 15
Back to Lesson 13
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