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File: MM_STD_07

!!! MM_STD Tutorial part 7

[[[> images/MM_STD.png ]]]
Here is the diagram from the end of part 6.  Remember, the
missing arrows have been left out deliberately because they
correspond to transitions that are, in some sense, less
fluid, less natural.

Now it's time for the payoff. Since this series started with
a question about the TimeReversal of MillsMess, let's
start our analyses with that.

We need to start by working out what MillsMess is on the
diagram.  On the way we'll see some of the limitations of
the diagram, as well as the basic power of the technique.

MillsMess.  I'll start the description and analysis from
where you have the left arm crossed over the right, and
you're about to do an exchange with the left hand.  This
throw therefore comes from the right side.  Here's a full
description of the sequence of throws:

All catches are made more-or-less in the middle of
JuggleSpace. Remember that, and we'll come back to it
later.

So, what does all this mean on the diagram?  We can easily
write down what the states are, so let's add them to the
chart above ...

[[[< images/MillsMess0.png ]]]
So here is the diagram with the states and arrows.
This gives us the transitions
between the various states, and
as a simple consequence it also
tells us some exchange types:
  * Throw 1 - inside  exchange
  * Throw 2 - outside exchange
  * Throw 3 - unknown exchange
  * Throw 4 - inside  exchange
  * Throw 5 - outside exchange
  * Throw 6 - unknown exchange

We know what type of exchanges occur on throws 1, 2, 4 and 5
because only one type is permitted by the state transition
diagram.  But what about the unknown exchanges?

Throw 3 is with the left hand from the right side of the
body and the catch is in the middle of JuggleSpace.  This
means that between the throw and catch of the exchange the
hand moves to the left. That makes it an inside exchange.
Let's put that in, along with all the other transition
types, just for the sake of completeness.

[[[> images/MillsMess.png ]]]
So what was the point above about the catches?

Well, at the very beginning we assumed that the exchange
happened more-or-less in a single location. This isn't
true in MillsMess. When going from  *Ll*  to  *Ur*  with
our inside exchange, the catch happens when the arms have
uncrossed.  This contrasts with the assumption that the
throw and catch happen, and then the hands move to their
new places.

However, the notation doesn't really cover the actual hand
positions.  This is one of its major limitations, although
there are ways of getting around it.  In fact, when you're
juggling, your DwellTime tends not to be extremely long.
This means that, in general, the catch happens close to
the position specified by the _next_ state, not this one.

A specific example is the  *Ll* to *Ur*  transition.  If we
did have the long DwellTime then the catch would happen
with the arms still crossed.  However, for MillsMess the
catch happens in the uncrossed position, as specified by
the next state.

We will assume from here on that this is, in fact, the case.
We'll assume that the catch happens close to the position
specified by the next state.

So, MillsMess is a clockwise traversal of the diagram,
following the "inside exchange" arrows whenever you have
a choice.  And the TimeReversal ??   That's the subject
of the next page.

!! Next

MM_STD_08