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How To Write Down A Juggling Pattern:

A Guide For The Perplexed

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The contents of this document are Copyright (C) Solipsys Ltd, 1996, but you may reproduce and redistribute them freely provided that you make no changes, no charge, that this copyright notice remains attached, and you provide a link to the original.

A Word Of Warning: You're sitting in front of a computer, right? And you're about to learn something about juggling. You may even feel the need to do some. Now you know this might not be a good idea. It could get expensive. We'd just like to make it plain that if you break anything, it's not our fault. We warned you, OK?

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Site Swaps are a notation for writing down juggling patterns. This document is a gentle introduction, assuming that you already know how to juggle. If you don't already know how to juggle but would like to learn, you might care to look at the juggling animation and tutorial program Juggle Krazy, or at some of the other web pages we've listed under Further Reading.

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Contents

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Introduction

This is intended to be an extremely gentle introduction to Site Swaps. You may think it's a bit slow in places, but you can always skip ahead and then come back if you think you've missed something.

For this entire document we're going to assume that we're juggling balls. In principle at least, everything we do with balls we can also do with clubs and rings, so we may as well stick to balls for now. We're also not going to deal with multiplex or synchronous patterns. Everything here will assume that each hand holds, throws and catches at most one ball at a time, and that the hands always take it in turns.

Another thing we're going to ignore, along with clubs, rings, flaming torches and chainsaws, is hand movements. We're going to pretend that the hands always stay on their own side of the body, and that apart from the small movements needed to actually throw and catch, we'll pretend that they don't move around at all. Of course, this means that we're ignoring most of the patterns that any performer (or the public, for that matter!) will think are interesting. It means we won't allow Mills Mess, Burke's Barrage, Rubenstein's Revenge, behind the back throws, or any of the carry type variations. The Site Swap notation by itself doesn't cover these sorts of things.

Finally, we're only going to deal with exactly two hands. We expect that documents covering hand movements, multiplex, multi-hand patterns and passing will be developed soon.

Questions and comments on this tutorial/explanation would be very welcome. We want to improve this document so as to make it better and better. Your assistance will be greatly appreciated.

Besides, let's be honest. We'd like to sell you something. A good way of getting to grips with the Site Swap notation is to use our juggling animation package Juggle Krazy. You can certainly use the notation without any computer assistance at all, however, so we won't mention Juggle Krazy any more in this document. Err, well, not much anyway.

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Lesson 1: The Basic Rhythm

We start by pretending that we're juggling in time with some music. Suppose that it's a nice comfortable rhythm for doing an ordinary three ball cascade. Every person has their own speed, so imagine your own music.

Notice that each ball is thrown in turn.

	Red, green, blue, red, green, blue, red,
	green, blue, red, green, blue, red, green,
	blue, red, green, blue, red, green, blue,
	etc.
Now, if you have to juggle a four ball fountain to this same music you will have to throw higher. Obvious, yes, but important. When juggling to a constant rhythm, more balls means higher throws. In fact, most people juggle slightly faster as they increase the number of balls, but for this discussion we need to pretend that we always juggle to the same piece of music.

A technical point here. We're going to ignore the actual physical heights the balls go to. For this discussion it's not important. It is very technical, and if you're really interested, it is covered below in technical notes [2].

So, here we are:

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Lesson 2: Types of Throw

Remember our notes from Lesson 1:

Now, it turns out that if you juggle all the balls to the same height, always alternate hands, and juggle at a steady rhythm, then even numbers must be done in a fountain, and odd numbers must be done in a cascade. The reason for this is explained in technical note [3] below, but feel free to read it later (or never!) if you either know this already or are prepared to accept it as given for now.

Now, this next bit is going to seem obvious. We have already seen that if we always juggle to the same piece of music, the different patterns go to different physical heights. We are going to represent each height of throw by a number. The height at which we juggle four, we'll call that "4". The height at which we juggle 7 (we wish!) we'll call "7".

(You can safely skip this next paragraph if you want, but only one!)

It's very important to know that the physical height a ball goes to is not proportional to the number that we are using to represent the throw. An "8" does not go twice as high as a "4". The number is not the height, it represents the type of throw. For more details, see technical note [2].

This may all seem a little strange, but if you think of a "4" as representing the kind of throw you do when you juggle four balls, you won't go wrong.

Now, and you'll feel like we're repeating ourselves here, a juggling pattern can simply be represented by a sequence of numbers that tell you the kinds of throws. For a four ball fountain it's obvious, they're all 4's! So a four ball fountain is simply

 
          ... 4 4 4 4 4 4 4 4 4 4 ...

Similarly, a five ball cascade is simply

 
        ... 5 5 5 5 5 5 5 5 5 5 ...

and so on.

So, to summarise:

and now

In the next lesson we'll see that the numbers can be mixed up - they don't all have to be the same!

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Lesson 3: Variations

Just to remind you:

Let's now take a closer look at what happens when we juggle four balls in a fountain. Remember, each ball gets thrown in turn, so whenever a ball is thrown it is then thrown again four beats later. But what if, without warning or provocation, in the middle of a four ball fountain, we throw one of the balls as if it were in a five ball cascade?

Well, three things to note.

  1. It will change hands.
  2. On landing, it will clash with the ball that is thrown immediately after it.
  3. It will leave a hole where it used to land.

Have a look at points 2 and 3. The ball we have just thrown as a 5, high and crossing, is going to collide with the next ball's landing, and leave a hole behind. The obvious thing to do, then, is to throw that next ball into the hole that's been left behind. This will both avoid the collision and fill the gap. So what kind of throw will that be, then?

Well, it will cross, and it has to come down earlier than it used to. In fact, it turns out to be a 3! The two balls involved trade landing places. The first one thrown lands when the second one used to land, and the second one comes down early to land when the first one used to. The first comes down a beat late, making it a 5 instead of a 4, and the second comes down a beat early, making it a 3 instead of a 4. The pattern can therefore be described as

 
        ... 4 4 4 4 4 4 5 3 4 4 4 4 4 4 ...

In fact, if you get the shareware version of Juggle Krazy and put

 
        4 4 4 4 4 4 5 3

in the text window (note, exactly six lots of 4, and spaces between the numbers) and hit the "Accept" button, you'll see two balls continually swapping places in a four ball fountain.

You can try the same thing with fewer 4's. The only problem is that the swapping of balls happens more often, and doesn't always involve the same two balls. This makes it somewhat harder to see exactly what's going on, but it still works.

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Lesson 4: Some Examples

Again the reminder:

OK. Remember we had this ...

 
        4 4 4 4 4 4 5 3

As we said above, the exact number of 4's doesn't really matter. We put six of them in so the swapping is of the same balls every time (yellow and green in the case of Juggle Krazy) and so that it happens at a relaxed rate. As has already been commented on, however, the number of 4's can be changed. Each of the following works...

(We'll occasionally leave out the spaces for brevity from now on.)

44444453 we have already seen. 4444453 is just a little bit faster. 4453 has the continual exchange of two specific balls running as fast as it will go. Any fewer 4's and it won't be the same two balls that keep swapping every time. 453 is particularly interesting. The "3" turns out to be done with the same ball every time.

The last pattern in this list, 53, is also special. In this pattern the right hand always does 5's. Always. The right hand, in fact, has no idea that it isn't doing 5 balls. Well, except for the fact that the incoming throws are all coming in low and lobbed instead of descending from a great height. The left hand always does the low throw, and so the left hand thinks it's only doing three balls.

Here are some more examples. Ignore any collisions for now, we'll worry about getting rid of them another time.

(optional) See if you can spot a simple rule that tells you how many balls are in the pattern.

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Lesson 5: Holds, Transfers and Empty Hands

Now, this lesson is almost (but not quite!) completely optional, and this first bit is all you really need to know, although the reasons are interesting. Still ...

We know what 3's, 4's, 5's and so on mean. What about 2's, 1's and 0's?

It turns out, although the reasons are a little more complicated, that we interpret these as follows.

That's all you actually need to know, and you can now safely skip to the next lesson any time you want to. We'll still go gently, though, because some people find this bit interesting too.

So, if you want to know more, here are the reasons why. You'll be getting tired of seeing this now, but for this lesson it is particularly important ...

Now let's go back a little and recall what the numbers mean. We say that a "4" is the kind of throw in a four ball fountain, but why?

The idea was that the ball will next be thrown four beats later. That is what the number for a throw means -how many beats later will the ball next be thrown, regardless of the hand. In a three ball cascade each ball gets thrown every third beat, and in a four ball fountain each ball gets thrown every fourth beat. So whenever you write down a number it tells you how many beats the ball will take before it next gets thrown.

So what about a 2? Well, the 2 means that the ball must next be thrown two beats later, and that will be the same hand again. We can say more than that, though. Since nothing will have happened in that hand between the throw and catch you may as well hold the ball. You can throw it if you like, but it doesn't have much time to get out of the hand, so it won't go very far. You may as well hold onto it.

Try having a look at

 
        4 4 4 4 4 4 5 5 2

Note - exactly six lots of 4 again. This is a four ball fountain with a two high flash. The high throws are the 5's, and the 2 is holding a ball while waiting for the others to come down again. You could do a little bob with the 2 if you like, but it's difficult and pretty pointless.

What about a 1? The 1 means that the ball must next be thrown on the very next beat. So if you do a 1 with the left hand, the right hand must throw that same ball on its next throw. That means that the ball must be handed across and not waste any time in the air. It must be a transfer.

And how about a 0? Have a look at the following. They are all possible juggling tricks with 4 balls.

           ... 4 4 4  4  4 4 4 ...
          ... 4 4 4  5 3  4 4 4 ...
         ... 4 4 4  5 5 2  4 4 4 ...
        ... 4 4 4  5 5 5 1  4 4 4 ...

Can you see what comes next? With the eye of faith, you may agree that the pattern is completed like this ...

           ... 4 4 4  4  4 4 4 ...
          ... 4 4 4  5 3  4 4 4 ...
         ... 4 4 4  5 5 2  4 4 4 ...
        ... 4 4 4  5 5 5 1  4 4 4 ...
       ... 4 4 4  5 5 5 5 0  4 4 4 ...

This means that we should be able to juggle this last pattern. But what is it?

Looking at the numbers we can see that we're doing a four ball fountain. Then we throw four high, crossing throws. In essence we do a four high flash out of a fountain, momentarily doing as much of a five ball cascade as we can. But now we have no balls left, and by our rule that we make a throw on every beat of the music we have to make some sort of throw, even though we have nothing left to throw. So we say that it's a "0", and that's an empty hand for one beat.

There are two other ways of thinking about a "0", neither of which is very useful, but both of which are very entertaining to technical types. These are mentioned in technical note [4].

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Lesson 6: Some More Examples

Here are a few more examples along with their very much more verbose English versions ...

Be warned, simply reading these will not necessarily be very enlightening. The best thing to do is run them in super slow motion in Juggle Krazy and compare it with the description ...

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Lesson 7: Inventing Tricks

OK, once again, the reminder:

By now we hope you can interpret the lists of numbers, so here are some particularly nice ones for you to try ...

Three balls ...

Four balls ...

There are lots more, and now we'll show you how to invent your own. It's easy once you know just one simple trick, although there are several ways of actually going about it. We'll show you our favourite first.

Firstly, choose how long you want the pattern to last. This can be anything from just one throw (boring) to 100 throws (impossible to remember!) For this example we'll choose 5. Remember, this is not the number of balls you're going to be juggling. It's how long the sequence of throws will last. Write down a row of that many (5) x's, each one with a dot underneath - like this.

 
        x x x x x
        . . . . .

What we're going to do is to replace each "x" with a number representing a throw. This can be any number from zero on upwards, but if you put 100 then you'd better work on your high throws! Let's start by putting a 4 in the first place.

 
        4 x x x x
        . . . . .

As you know, this throw is the kind of throw we do when juggling 4 balls in a fountain, so, in particular, it's next going to be thrown four beats later. Count forwards four places, and replace the dot with a "*". Like this.

 
        4 x x x x
        . . . . *

Now we do it again. Replace any "x" with a number, count forwards, and put a "*" on the dot. The only thing is that you're not allowed to put an "*" down unless there's a dot there. For example, you're not allowed to put a 3 in place of the next "x". Let's try a 6 instead. This gives ...

 
        4 6 x x x
        . . * . *

As you count forwards here you fall off the end, so come back on at the beginning and keep counting. The "*" ends up just one place further on from the 6, even though we've counted six places.

Now let's put in another number.

 
        4 6 4 x x
        . * * . *

We fell off the end again, and again we came back on at the front. Things are getting tight now. The fourth "x" can't be replaced by a 1, 3, 4, 6, 8, 9, etc, so we can pick 0, 2, or any multiple of 5 added to these. Try it. Let's use 5.

 
        4 6 4 5 x
        . * * * *

The easy choice is now a 1, and we get this.

 
        4 6 4 5 1
        * * * * *

Finished. Put it into Juggle Krazy and see what you get!

This system can be used to invent any number of repeating Site Swap patterns. Try it for yourself, but remember that the larger the numbers, the harder the pattern. There are other things that make a pattern hard too, but that's still not fully understood. In general you simply have to try a pattern for yourself and see if you can do it.

See also Inventing Synchronous Patterns.

In the next lesson we'll see how it can be used to work out transitions between Site Swap sequences.

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Lesson 8: Transitions

So let's suppose you're doing a three ball cascade and you want to move efficiently into a three ball shower, which in Site Swap is "5 1". Write down the sequences...

 
        . . . 3 3 3 3 3 3 x x 5 1 5 1 5 1 5 1 . . .
        . . . . . . . . . . . . . . . . . . . . . .

From each number count forwards, just as before, and replace the dot with a "*" ...

 
        . . . 3 3 3 3 3 3 x x 5 1 5 1 5 1 5 1 . . .
        . . . . . . * * * * * * . * . * * * * * * . . . .

You'll see that there are two numbers missing and two gaps in the second row. Each "x" can go to either dot. Taking the possibilities in the non-obvious order, send the first "x" to the second dot so you replace the first "x" with "5". The second "x" then has to be replaced with "2", giving the sequence

 
        . . . 3 3 3 3 3 3 5 2 5 1 5 1 5 1 5 1 . . .

This means that from the three ball cascade you throw the first ball from the shower, wait, and then do the shower. During the pause you can do a pirouette or something, particularly if you make the pause longer, like this ...

 
        . . . 3 3 3 3 3 3 7 2 2 2 5 1 5 1 5 1 5 1 . . .

This is the transition most people use to go from cascade to shower.

The other possibility is to replace the first "x" with "3", sending it to the first dot, and replace the second "x" with a "4", sending it to the second dot. This gives

 
        . . . 3 3 3 3 3 3 3 4 5 1 5 1 5 1 5 1 . . .

This gives a transition that deserves to be more widely known. It goes like this. Do the three ball cascade. Throw a single fountain throw from the left hand, followed immediately by the shower from the right. The shower is suddenly there with no preceding pause. Great for confusing jugglers who don't know about it.

You can use this technique for finding transitions out of a shower too, or between any two Site Swaps.

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Conclusion

Have you got this far? Congratulations!

Now, what would you like to see here? There's a lot more about Site Swaps that we could tell you, but at this point we would like to know what you would like us to explain.

How to Contact Us.

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Technical notes

Note 1: Dwell Time

This has been moved to the Dwell Time page.

Note 2: Physical throw heights

Before reading this technical note you need to read about the Dwell Time.

Suppose the Dwell Time is D, usually around 0.7 or so, and suppose the time between beats is T. Then for a Site Swap value of V the actual time from throw to catch is (V-2*D)*T. (We multiply the Dwell Time D by 2 because it's two beats between successive throws from the same hand.) Call this quantity Z.

We note that if the Dwell Time is close to its usual value of 0.7 then this only works for values of 2 or more.

Remembering that the ball has to go both up and down, so half the time is on the journey down, the physical height H is given by the formula

 
        H = 1/2 G (Z/2)^2
          = 1/2 G Z^2/4
          = G*Z*Z/8

where G is acceleration due to gravity, roughly 9.8 metres per second squared. As a quick reality check, suppose you do about 4 throws/catches per second with 5 balls, and have a Dwell Time of 0.7. Then

 
        Z = (V-2*D)*T
          = (5-1.4)*0.25
          = 0.9

Then

 
        H = 9.8*0.9*0.9/8
          = 0.992250 metres.

or just over three feet, just about right.

Actual and ratios of physical heights of different throws are given in the following table.

SiteSwap
Value
Height
in metres
Height compared
with 3
-------- --------- ---------------
3 0.20 1.00
4 0.52 2.64
5 1.00 5.06
6 1.62 8.27
7 2.40 12.25

As you can see, four is about two and a half times as high as 3, 5 is about twice as high as 4 or five times as high as 3, and 6 is about three times as high as 4.

Note 3: Cascade versus Fountain

If the balls are always going to the same height then they must take it in turns. No ball has a chance to overtake another, so there is no other choice. Also, if you're juggling four balls then every fourth throw will be the same ball. But every fourth throw is with the same hand, assuming the hands always take it in turns, so that means that if you're doing four balls, and they all go to the same height, then each ball must come back to the same hand. You have to do a fountain.

The same reason goes through for any even number of balls, and identical reasoning works for odd numbers of balls too, only now the balls have to swap hands, because every (say) fifth throw is with the other hand.

Note 4: Empty hands

Suppose that the Dwell Time is 50%, so that each hand is full for exactly half the time. This means that for every throw each ball will spend one beat in the hand, and the rest of the time in the air. A "5" will spend four beats in the air and one in the hand before being thrown again, a "4" will spend three beats in the air and one in the hand. Carrying on with this trend, a "1" must spend no time at all in the air. But what about a "0"? It must spend -1 beats in the air. That means that the ball when thrown must go backwards in time by one beat!

There are various interpretations of this, and an entire thesis can be written on it. However, one way of looking at it is that the "catch" is actually the creation of a ball/anti-ball pair. The ball is in the hand, the anti-ball is in fact the ball going backwards in time. Then, one beat later, the ball and anti-ball re-combine. The hand once again becomes empty as the two mutually annihilate.

But what about the number of balls? If we do 55550 in the middle of a four ball fountain there are four balls in the air. What about the ball from the ball/anti-ball pair? Doesn't that make 5 balls?

No, because the anti-ball is a negative ball, and the balance is retained.

Although this all sounds bizarre it in fact has precise analogies in modern physics in which anti-particles are treated as being identical to particles moving backwards in time.

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Further Reading

This list is far from exhaustive (and must be expanded!), but you can find plenty more places to look by visiting the above addresses.

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How To Contact Us

Send comments or questions to Colin Wright: mailto:SiteSwap@solipsys.co.uk

By snailmail, we can be contacted at:

Solipsys Ltd.,
4 Brook Street,
Port Sunlight Village,
Wirral, CH62 5DB,
England


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