In 1985 there arose, simultaneously in three places around the world, by groups entirely unconnected and completely ignorant of each others' existence, a notation for juggling tricks. The notation was incomplete, since not every trick could be described, and like many notations, it was not immediately apparent to the uninitiated how to read it, how to use it, or whether it would be of any real use. For those who understood it, however, it was instantly obvious that it was right. Somehow the notation managed to capture the essence of those tricks it described, and the fact that the same notation arose in more than one place at once showed that its time had come, and it was, quite simply, the notation.
Since then the notation, now known as Site Swap notation, has become fairly well-known in juggling circles. Reactions were initially somewhat mixed. In juggling, as with music, there are those who study the works produced by others, there are those who produce their own, and there are those, the juggling equivalent of the jazz musician, who feel that while it may be of use to some, juggling has a soul, and should not be trapped, caged, and prevented from varying from one, single form. Over time, however, the notation has gained acceptance by the majority, and it is now considered a useful tool for the communication of tricks, incomplete though it is.
It is impossible to show in written form the infinite variety of juggling tricks that can be performed. Some have the arms moving sinuously past each other, somehow managing to toss, catch and carry three balls, never more than one per hand at a time, always moving over and past each other. Others have the hands largely stationary with the balls, rings, clubs, fire-torches or chainsaws spinning to various heights, seemingly none the same. Such variety can never fully be captured, and there is always room for the performer's own interpretation of the basic moves, the underlying patterns. The Site Swap notation describes the trick that is the basis on which variations can then be built.
To make precise the limitations we place on ourselves, we state a specific set of rules that must be obeyed. These rules seem terribly restrictive, but within the resultant framework we will find that there is structure that can be exploited. The rules we use are as follows:
To investigate the consequences of the rules, let's look at juggling three balls, and let's start with a further simplification. For now, let's assume for now that we not only obey the rules, but also insist that every throw must be identical.
Some quick experimentation soon shows that this then forces each ball to be thrown in sequence, for otherwise there would be variations in the timings. We start by throwing ball A with the right hand, then ball B with the left, then ball C with the right. Now we have to make a throw with the left hand and it must be ball A. Each ball is thrown every third throw, and from this we can conclude three things. The first is that it will work, the second is that the balls don't go in the expected "Big Circle" seen on every popular picture of a clown juggling, and the third inescapable conclusion is that the balls are forced to change hands.
Do the same thing with four balls and we discover now that the balls cannot change hands. If we juggle four balls with every throw the same, and subject to our rules, we end up juggling two balls in each hand, independently, but asynchronously. There is no alternative, and in the live version of this presentation, whether it is to adults or to children (generally 14 or older), to scientists, mathematicians or lay people, the audience can see this for themselves.
With five balls they once again cross, with six balls again it becomes half in each hand. These are unavoidable consequences of our rules. With each ball being thrown in exactly the same way, each ball is forced to take it's turn. With an odd number, that means that each ball will alternate hands, but with an even, each ball will have to come back to the same hand.
To make it easier to examine what's happening in a juggling pattern we make use of a diagram such as Figure 1. In this diagram the vertical grey lines represent the hands, the horizontal grey lines show the beats of the music to which we are juggling, and the coloured lines bouncing back and forth between the hands represent the balls. Figure 1 shows the three ball cascade, the pattern with three balls in which every throw is the same.
We can see clearly now how the balls weave around each other. Indeed, if a ribbon is attached to each ball, and the other ends of the ribbons held some distance away, juggling this pattern will plait the balls.
Diagrams such as Figure 1 are called Space-Time Diagrams because they show both space and time on the same chart. Jugglers will often call it a Ladder Diagram, for obvious reasons.
If we look at how long each ball spends in the air, an obvious pattern emerges. With three balls, each ball is thrown every third throw, and since it spends one beat of time in the hand it must therefore spend two beats of time in the air. Similar reasoning gives us the following table:
|... and so on ...|
The central column here is called the "Cycle Time," and turns out to be the single most important idea in the development of the theory of juggling.
From this table we can, should we choose, deduce what the physical throw heights must be and derive formulae for how hard we must throw. This, however, is not the most interesting direction to take.
Consider now what happens if we are juggling four balls in the pattern where all of the throws are the same. Every ball goes into the air, comes down, gets held in the hand, and then goes off into the air again. If any single throw is too high, or too low, the pattern is ruined.
However, if one single throw is made as if for one
instant we were juggling five, and the very next throw
is made as if we were juggling only three, what happens?
The first throw is higher, stays in the air for a beat
longer, and comes down in the other hand. We would
expect it to clash with another ball landing at that
time in that hand, but that has been preempted. The ball
with which it would have collided is the very ball that
has been thrown low and to the other hand. The timing
has conspired to allow the two balls to exchange places
in the pattern. They have swapped their landing sites,
and hence the name for the notation, Site Swap.
This rather unlikely pattern is annotated by writing down the cycle times of the throws involved: ... 4 4 4 5 3 4 4 4 ..., showing that in the middle of an infinite sequence of throws with cycle time "4", we do a "5" followed immediately by a "3".
Click SS:44445344 to see this trick.
There are other possibilities too. Here is a table of such sequences.
a: ... 4 4 4 4 4 4 4 ...
b: ... 4 4 4 5 3 4 4 4 ...
c: ... 4 4 4 5 5 2 4 4 4 ...
d: ... 4 4 4 5 5 5 1 4 4 4 ...
The fourth of these sequences, ... 4 4 4 5 5 5 1 4 4 4 ... was, incidentally, discovered by writing down exactly this sequence of sequences. The first three were already well-known to the juggling community, and given the clear and obvious progression, the fourth really ought to be a juggling trick, and it is.
But what does it mean to have a cycle time of 1? By referring to our table above we can see that the air time is always one less, so that implies that a throw with a cycle time of "1" should have an air time of "0". Does that make sense?
Yes it does. Try to "juggle" ("manipulate" might be a better word) only one ball so that each hand is full for exactly half the time. The ball must effectively teleport between hands, and we do this simply by passing the ball from one hand to another. We cheat slightly, technically we are breaking rule 6, but in principle it can (almost) be done.
What does pattern "d" look like? In the middle of juggling two balls in each hand we must suddenly launch one high and crossing, and the next high and crossing, and the next high and crossing. For a moment we are effectively juggling 5 balls, but it can't go on. We only have one ball left, and we want to make two more throws. So we cheat, and slide the remaining ball directly into the other hand. The other three come down, and we are once again juggling 4 balls in the standard pattern.
You can click on these links to see this trick:
What about the obvious missing sequence above, why stop at sequence "d"? Surely we should be able to juggle ... 4 4 4 5 5 5 5 0 4 4 4 ... ??
Pause for thought: what does a cycle time of 0 imply? It requires that the ball have an air-time that is one less, which is -1. It would require that when the cycle time 0 throw is made, the ball must travel backwards in time for one beat.
Unbelievably, this can be done! There are three interpretations. One is purely practical, and the other two are straight out of the fairy-tale world of quantum physics.
The most theoretically satisfying explanation is that the ball really does go backwards in time. During the live version of this presentation we will have met the idea of a space-time diagram and used it in depth to analyse several juggling tricks. The most obvious space-time diagram for the sequence ... 4 4 4 5 5 5 5 0 4 4 4 ... is shown in Figure 3.
As we can clearly see, the cycle-time 0 throw goes backwards! Of course, many would argue that this simply isn't physically possible, and in some senses they would be right, but there's more going on here than that.
How would a flash from a camera be represented on the
space-time diagram? It would be everywhere in space, but
only occur at one point in time. It would therefore be a
horizontal line on our diagram, and by seeing where the
trace from each ball crosses that line we can see where
each ball would be in the photograph. In Figure 4a we
have drawn a horizontal line at the time of a flash, and
we can see that there's a ball in the right hand, a ball
in the air above the right hand, a ball coming in to the
left hand, and the first of our high throws has just
left the left hand on its way to the right hand. In
Figure 4b there is a high throw just leaving the left
hand, a ball that's just landed in the right hand, and
two high in the air crossing from hand to hand.
In Figure 4c we have two high in the air crossing over,
high throw number three has just left the left hand, and
the right hand is holding the last ball to be thrown. In
each case we have four balls.
What about Figure 4d ??
There are four balls high in the air crossing over, and one ball in the left hand. That makes a total of five balls so far when we are only juggling four!
The other ball, the one travelling backwards in time, must therefore be a negative ball. This is exactly what quantum physics says. In all the quantum physics equations an anti-particle is identical to a particle travelling backwards in time. A positron is simply an electron going backwards in time, a proton is an anti-proton going backwards in time, and a photon is its own anti-particle so it doesn't know if it's coming or going, and that's closely related to why it travels at the speed of light, and that ties in with the Special Theory of Relativity. So what we have at time X is the mutual creation of a ball/anti-ball pair, and at time Y we have the mutual annihilation of the same pair. [See footnote 2]
Of course the truth is much, much simpler.
The rules we laid down gave us a model of juggling, and we have been working inside the model. Until now it has agreed closely with reality, but we have finally broken some of the rules. We are juggling four balls and have made four high throws. There are no balls left, and so we cannot have our hands full for half of the time. The theory insists on it, and thus creates this anti-ball, but the reality is that for one beat we have an empty hand. What else can juggling 0 balls be but standing there with empty hands.
We can use this to draw a clear distinction between models and reality, and to show that an inaccurate model is not a barrier to usefulness. Provided we understand the assumptions on which our models are based, and recognise the inappropriate predictions caused by our stepping outside them, the models can still be useful.
As a case in point we have used our model to create a method of inventing juggling tricks. Not all sequences of numbers are valid, legal juggling tricks. This can form the basis of an investigation: which sequences work, which ones don't, and why? Using the space-time diagram can assist in the understanding of the ideas, but using an animation package helps even more.
There are several things to spot about this table. Just one example: the numbers in the final sequence always average to the number of balls in the juggling trick. It works even if you start with negative numbers.
The real questions are:
Here's another way to invent juggling sequences.
Start anywhere, follow the arrows, write down the black numbers as you go, and finish where you started. The result is always a juggling sequence.
In closing, it's worth reviewing the processes that take place during the presentation. Most of the audience cannot juggle, and will never have taken a close look at juggling, and yet along the way they deduce that the balls must cross when there's an odd number of them in the standard pattern, and cannot cross when there's an even number in the standard pattern. They deduce the consequences of the rules, use them to invent juggling tricks, they see what happens when the model is pushed too far. They observe, form hypotheses and test them. In short, they are following the idealised scientific process. By three-quarters of the way through they have seen the distinction between models and the reality they represent, recognising that models are intended to help us understand the reality. They have also have seen what happens when things go wrong!
Towards the end of the presentation I generally find that several of the audience are asking "Why?" and "Does it always work?" which gives me the ideal opportunity to investigate justification and its close friend - proof. The power of a proof is that it can show that something must happen, can never go wrong, no matter how many examples you try. It lets me draw the distinction between proofs that are enough for everyday use and proofs that allow for the most bizarre situations. Thus we can see how mathematical proofs relate to everyday proofs; their similarities and their differences.
In short, this presentation gives literally dozens of starting points for investigations, proofs, projects and understanding. More than that, it shows that mathematics isn't just arithmetic, and isn't only in the classroom. It emerges in the most unlikely places.
Juggling, for example.
(c) Colin Wright,
Footnote 1: The Juggling Information Service can be found at http://www.juggling.org and contains a very large repository of information about juggling and related material. Unfortunately it is not currently being maintained, but one can find several juggling animation packages there. Another resource is here:
Footnote 2: The purists will argue that the creation of a ball/anti-ball pair must require enormous amounts of energy. However, just as there is an uncertainty principle linking momentum and position, so there is an uncertainty principle linking energy and time. Since we are assuming that the throws and catches occur at specific, and therefore known, times, we cannot know how much energy is involved. Therefore we can borrow from the quantum uncertainty for energy and create a virtual juggling ball/anti-ball pair, provided we have them annihilate within an appropriate time frame. The anti-ball cannot exist for very long. This is a direct analogy with the quantum vacuum.
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