From: celarier@reed.edu (Stuart Celarier)
Newsgroups: rec.juggling
Subject: Site Swap Notation and the Three Ball Half Shower
Date: 14 Oct 92 20:48:25 GMT
Article-I.D.: reed.1992Oct14.204825.29926
Organization: Reed College, Portland, OR

In article <1992Oct8.191711.21616@nntpd.lkg.dec.com> miller@quiver.enet.dec.com  
(Brian A. Miller) writes:
> ...
> Back to the original question of half showers:  a 3 ball half shower is (in
> my mind) still basically a "3" (not a "53"), though the throws from one hand
> go slightly higher than the throws of the other hand.  Can we agree on this
> point first, and then start talking about 4 balls again with a better grasp
> of terminology?
> 
> --BAM
> Brian A. Miller


Brian is certainly not the first person to have this problem, so let me see if  
I can help straighten it out.

	THE THREE BALL HALF SHOWER CANNOT BE COMPLETELY DESCRIBED
	USING SITE SWAP NOTATION.

There, I said it, now let me see if I can rationalize my way out of this one.

First, let's agree on what site swap notation (SSN) is and isn't.  SSN is a  
system for expressing a certain class of tricks.  These tricks are all  
characterized by each hand throwing alternately and all throws made at a  
constant speed.  Each throw is described by a single number (nonnegative  
integer) which indicates only two things:  how high the toss is, based on the  
magnitude of the number; and which hand the toss goes to, based on the parity  
of the number.

The numbers represent the kind of throw one would have to make to juggle that  
many objects (in two hands) at a constant speed.  As the inventors of SSN,  
Bengt (may he rest in peace) and Boppo explained, when they first presented SSN  
at the IJA Convention at UCLA, for odd numbers, the pattern is based on the  
cascade of that many objects, while for even numbers the pattern is based on an  
outward shower in each hand of half the objects.  Additionally, to keep the  
throws at a constant speed, a 5 has to be thrown a good deal higher than the  
average 5 ball shower toss:  most people cascade five balls (that's 5 in SSN)  
faster than they cascade three balls (3 in SSN).  So to make the timing of SSN  
tricks work, either the 5s will have to be higher or the 3s lower.  Similarly a  
7 should be very high indeed.

So much for the ground rules:  the rest of the theory of site swap notation  
lies in finding which sequences of numbers represent valid SSN tricks, for  
example a trick cannot require you to catch to balls at the same time, nor can  
it require you to throw a ball from an empty hand.  And there is some very  
interesting mathematics that goes along with determining the valid SSN tricks.

And that is ALL there is to SSN.  The notation does not take into consideration  
hand positions or crossings, nor the "dwell time" (i.e., the amount of time the  
ball is held in your hand).  The notation does not account for different kinds  
of throws, or throwing and catching more than one ball at a time (multiplex).  
And the notation does not provide for tricks where the throws are at different  
rhythms, such as syncopated or even simultaneous throws.

So it is wrong to assume that all, or even most, patterns are actually  
legitimate SSN tricks.  (This was a major stumbling block for some at the UCLA  
talk.)  Even if a pattern appears to be made up entirely out of valid SSN-style  
throws, it is not necessarily a valid SSN trick:  it could have nonconstant  
throw times, or violate other "rules" of SSN.

This is not to say that SSN cannot (or has not) been extended to describe other  
tricks.  This is also not to say that site swap animation programs cannot be  
made to animate other tricks.  But those extensions or programs do not make  
something a SSN trick -- this is a big distinction.

As a matter of convenience and rigor, we want to say that a particular SSN  
trick is COMPLETELY described by the notation.  All other things which are  
outside of the SSN are outside of the notation, period.  Therefore, for  
example, there is one and only one pattern which is a SSN 3, and that is the 3  
ball cascade.

Now let me take on *why* a three ball half shower (3BHS) is certainly not 3 in  
SSN.  Some have suggested that the 3BHS is 3 because they can take their  
animation program and tell one hand to have a negative scoop (that is, one hand  
carries the ball away from the center prior to the toss).  SSN says nothing  
about scoop, and Bengt and Boppo implied, if anything, that all scoops are  
towards the inside of the pattern (cascade for odd numbers, one-handed outward  
showers for even numbers).

What's more, to achieve the visual effect of the half shower (as compared to a  
whole shower) one wants the balls thrown from the outside to go over, and hence  
higher than the balls thrown to the inside.  This is as Brian Miller correctly  
observes. If all other things are constant (including the dwell time) the  
higher ball must take longer (cf., Galileo Galilei).

Where we get into trouble is assuming that all seemingly-SSN tricks actually  
must be valid SSN tricks.  The notation and theory simply isn't that  
descriptive!  Brian states that the 3BHS is not  5 3  [sic, see below],  
therefore it must be the pattern 3.  This is the wrong conclusion.  The correct  
conclusion is therefore the 3BHS may not be a SSN trick at all, and I must  
examine all other 3 ball SSN tricks to see if there is one which describes the  
3BHS (there isn't one).

While the 3BHS is not a SSN trick, what the SSN suggests to us about the 3BHS  
is this.  If one hand is throwing higher, then the difference in time must be  
made up by a slightly longer dwell time in the other hand.  This means the  
pattern is syncopated.  I have carefully watched a lot of 3BHSs with this point  
in mind, and the amount of syncopation does indeed vary with the relative  
heights of the two tosses.

One early result from SSN theory is that average (arithmetic mean) of the  
numbers in a SSN trick must be an integer, and is in fact the number of objects  
being juggled.  Therefore,  5 3  is a four ball trick, not a three ball trick  
as Brian and others have suggested.  It is, indeed, a four ball half shower  
(4BHS).  Are there other 4BHSs?  Certainly, they involve changing the relative  
heights of the two tosses, which again results in varying degrees of  
syncopation -- nice patterns, but not SSN tricks.  In fact, many people learn  
the 4BHS with the hands throwing simultaneously.  Again, a wonderful pattern,  
and obviously not a SSN trick. 
--
Stuart Celarier		"No Problem, Easy Pickup" Portland Jugglers
celarier@reed.edu	PO Box 82842
503/331-1877		Portland, Oregon 97282