Subject: Hand-crossing patterns
From: Mike Day [View Profile]
Email: [members only]
Date: 17th Feb 1995.
This article explains a diagram which can be used to describe the
essential features of a number of juggling patterns, notably Mill's
Mess. It goes on to show how, when taken together with siteswaps, a
whole lot more patterns can be modelled, including Rubenstein's
Revenge. A method of extending siteswaps notation to incorporate the
information from the diagram is also demonstrated, and a simple
procedure is given for finding the time reversal of any pattern that
can be thus annotated.
I begin with a description of the diagram - I call it the Mill's Mess
State Transition Diagram, or MMSTD for short. I developed it around
1992, based on the idea that when juggling Mill's Mess (MM), although
one's hands pass through a continuum of positions, there are
essentially only a small number of definitely different states they
can be in. Either the right is crossed over the left, or the left is
crossed over the right, or they're not crossed at all. In addition,
for each of these positions, there are two possible hands which can
throw next.
Here's what it looks like...
Draw 6 circles in a 2-across 3-down formation, each big enough to fit
2 letters inside, and with a bit of spacing between circles so as to
allow connecting lines to be drawn. Label them Lr, Ll, Ul, Ur, Rr, Rl
(reading in the order you'd read a page of text). For each pair of
horizontally adjacent circles (ie 3 pairs), draw 4 connecting arrows,
2 going each way, with the arrowhead halfway along the line. For each
pair of vertically adjacent circles (ie 4 pairs), draw 2 such
connecting arrows, 1 going each way. For each vertical arrow going
away from circles with U's in (there are 4), put a bar on the point of
the arrowhead, like the symbol for a diode. Leave the other 4 vertical
arrows untouched. For each pair of horizontally connected circles, put
such a bar on 2 of the 4 connecting arrows, 1 going each way (leaving
the remaining 1 each way untouched). You now have MMSTD.
What does it all mean?
Each circle is a state you can be in. Take the U ones first - U
stands for 'uncrossed', and is the normal state for your hands. Each
U state is also given a lowercase letter, saying which hand is to
throw next. So a simple path which takes us back and forth between Ul
and Ur, following the non-barred arrows, represents a simple 3-ball
cascade (or indeed any siteswaps pattern with any number of balls,
but lets ignore that possibility for the moment!). The barred arrows
are for reverse throws - this follows the convention of the version of
siteswaps developed in winter 1984-5 at Cambridge University, England,
by Colin Wright, Adam Chalcraft, Jim Mellor and myself, in which a
reverse or outside throw is represented by putting a horizontal bar
over the number. So, following the barred arrows between Ul and Ur
describes reverse cascade. The remaining states are hands-crossed
states, with the capital letter showing which hand is on top. The
meaning of a reverse throw from a hands-crossed state is that it is
reversed with respect to the hand, not with respect to the body (the
hand is now on the other side of the body). eg a reverse throw from
state Ll (left hand on top, left to throw next) would involve throwing
the ball that's in the left hand to the left of the descending ball:
an ordinary throw from this state would have it thrown to the right
of
the descending ball.
It is important not to confuse an arrow going between states with a
ball that's being thrown. The arrow represents the abstract notion of
changing from one position of the hands (including which hand is the
next one to do something) to another such position. This change can be
thought of as happening instantaneously between the period of time
spent in the first state and the period of time spent in the second.
By contrast, each ball that's throws starts its flight at the moment
of a state change (or in fact just before), and can be in the air for
many state changes, depending on the pattern being juggled.
This model imposes restrictions on patterns. One such restriction is
that hands cannot be 'more crossed' than the amount they cross in MM.
For example, if you 'double-cross' your hands - cross your right hand
over your left and back under the left to the right again, and then
juggle from this position (I've seen it done!), then this would not
be modelled by MMSTD. Also, there are no arrows between eg Lr and Rl
- that is, hands are not allowed to uncross and cross again the other
way in a single transition. (Actually this can be modelled by the
diagram when taken together with an appropriate siteswaps pattern, eg
a 522 along (Lr,Ll,Ur,Rl) accomplishes the equivalent of a 3 from Lr
to Rl.) Finally, why are there only 2 arrows between vertically
connecting states, not 4? Well, this one's hard to explain. Basically
I have taken the view that it is the combination of the hand-state and
the position the ball is coming down that brings on the next state.
eg state Ul (hands uncrossed, left to throw next) and a ball coming
down to the right of my right hand, means I have to throw the ball
from the left (from where it is, otherwise it would be a different
state) and reach the left across, over or under the right hand, to
catch the ball. This is automatically a reverse throw, which is why
both arrows coming out of Ul are barred. Now obviously the ball could
have been coming down between the hands, or to the left of the left
hand, but then the exchange taking place in the left hand will not
lead to a hands-crossed state.
Now comes the good part...
MM is a clockwise circuit of all 6 states, following the non-barred
horizontal arrows. You would be advised at this point to verify that
this is the case.
Time-reversed MM is the same clockwise circuit, but following barred
horizontal arrows. This at first seems a little odd - why is it not
an anticlockwise circuit? If one draws a version of the diagram
containing only those arrows traversed during MM, and then applies
a time-reversing transformation to it, the reason becomes apparent:
in the time reversed version, there are 3 changes - all arrows go in
the opposite direction, so the cycle is now anti-clockwise; all
barred arrows become non-barred (and vice-versa), because a reverse
throw played backwards becomes a normal throw (and vice-versa); but
also the lowercase letter in each state changes to its opposite: if
the left hand is about to throw, then the right hand was the one to
throw last, so in the time reversal, the right hand is the one to
throw next. If the resulting picture is now redrawn so the states
are back in their normal places on the page, the cycle is now
clockwise and the only difference from normal MM is that the throws
between Lr and Ll and between Rl and Rr have become barred. This
points the way towards learning time-reversed MM: just make the
underarm throw go the other side of the descending ball. On that
basis, I have been able to learn reverse MM for a number of patterns,
including 4.
Hands-crossed cascades (and reverse cascades) can be got by moving
between Ll and Lr, or Rl and Rr, along appropriate arrows.
You might like to see what happens if you simply alternate between
eg states Ul and Lr. Or, if you take a circuit around just the top 4
states of the diagram, or the bottom 4. In fact I believe these
possiblilities may turn out to be useful exercises for the person
wishing to learn to juggle MM.
But what about an anticlockwise circuit of all 6 states? This turns
out to be a pattern like MM, but for each moment of MM where the
hands are crossed, the other hand is on top in this pattern - a sort
of 'upside-down' MM. (If you try to juggle this, take time to ensure
you're doing it right - it's not the same as a trick I think is
called the 'inverted mess'.) One question is - what favoured MM over
this pattern in terms of being discovered? Looking at the diagram,
there is no reason to favour one over the other. Juggling this
pattern certainly feels much less 'natural' than MM - it doesn't
seem to have the 'body logic' possessed by MM, though I can't quite
put my finger on the reason. Possibly it has something to do with the
throw that leads from an uncrossed to a crossed state - the throwing
hand is left with some upward velocity, and it would prefer to then
go over the top of the other hand, rather than change direction and
then go underneath.
The time reversal can be got by following the same procedure as for
MM.
What about siteswaps?
Of course any siteswaps pattern can be added to the state
transitions. eg do 4-height throws with the MM circuit, and you get
MM 4. You can also do things like 4-ball hands-crossed fountain,
MM 423, MM 534, etc.
Now, here's my claim:
Rubenstein's Revenge is the state-cycle
(Lr,Ll,Ur,Rl,Ur,Rl,Rr,Ul,Lr,Ul)
_ _
taken together with the pattern 52233, with the 5 corresponding to the
transition between Lr and Ll, etc. Or at least this describes a basic
pattern which can be modified to give Rubenstein's Revenge by
distorting the timing a little, so the 3's and the 5 become more
nearly the same height, and by putting in the characteristic
claw-type exchange. Again, you might like to take a bit of time out
to test this. (There is a bar over the 2 so as to remain consistent
with MMSTD - clearly this makes no real difference to a 2 throw.)
Developing notation...
If you do a siteswaps pattern in conjunction with a path around MMSTD,
the sequence of states can be represented by appending a suffix (A,B
or nothing) to each number of the pattern, and putting bars where
appropriate. This gives a nice compact representation of all such
tricks.
Here's how:
Any throw that takes you to a hands-crossed state, with the hand
that's just thrown now on top, is given a suffix of A ('Above').
Any throw that takes you to a hands-crossed state, with the hand
that's just thrown now on the bottom, is given a suffix of B
('Below').
Any throw that takes you to an uncrossed state is not given a suffix.
And that's all you need, because for any throw, the 3 aspects that
you would read off the diagram are now represented:
- whether the throw is l or r is determined by the parity of the
throw's position in the number sequence;
- the L,R or U aspect is given by the suffix on the throw (when taken
together with the l or r as already determined);
- the choice of barred or non-barred arrow (where there is a choice)
is determined by the presence or absence of a bar on the number.
Just as there are restrictions on the numbers in a siteswaps pattern,
so there are restrictions on the combinations of A's, B's and bars.
This follows from the restrictions imposed on transitions within
MMSTD,
as already explained. In particular:
(1) a subscripted throw following a non-subscripted one must have a
bar;
(2) a non-subscripted throw following a subscripted one mustn't have
a bar;
(3) an A-subscripted throw cannot follow an A-subscripted throw;
(4) a B-subscripted throw cannot follow a B-subscripted throw.
Under this scheme, I now list some patterns.
Where there is more than one way to 'Mill's Messify' a pattern, eg
534, I suggest the following convention. There is a natural 1-2-3 to
the 3 throws of MM, in terms of which most jugglers seem to think,
namely that the underarm throw is the third throw. (Perhaps you
disagree, in which case it would be easy to adopt a different
convention.) With this way of counting, in MM 534, the 4 is the
underarm throw. MM 345 is therefore a different pattern, and MM 453
different again.
_
Mill's Mess 3: 33 3
A B
_ _
Time-reversed Mill's Mess 3: 33 3
A B
_
'Upside down' Mill's Mess 3: 33 3
B A
_
Mill's Mess 4: 44 4
A B
_
Mill's Mess 345: 34 5
A B
_ _
Mill's Mess 42: 42 4 24 2
A B A B
_ _ _ _ _
Mill's Mess 12345: 12 3 45 1 23 4 51 2 34 5
A B A B A B A B A B
etc.
_
Burke's Barrage: 4 23 ie MM 234
B A
__
2-in-1 weave: 42 4 2
A B
__
Alternating weave: 42 5 22
A B
_ _
Rubenstein's Revenge: 5 22 33
B A A
_ _
Sean's Sequel: 4 23 33
B A A
(Sean's Sequel is a pattern discovered by Sean Gandini when learning
Rubenstein's Revenge.)
Once a pattern has been written down, its time-reverse can easily be
found. The procedure for doing this is as follows:
Call the pattern P, and its reverse R. Say P is of period n.
For 1<=i<=n, do the following 3 steps:
(1) If throw n-i+1 of pattern P is the destination of a throw of
height h, then throw i of pattern R is height h. (This is just the
standard rule for reversing a siteswaps pattern.)
(2) If throw n-i+1 of P is barred, then throw i of R is non-barred;
otherwise throw i of R is barred.
(3) If throw n-i (i