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(or mathematics of juggling)
Short history
Why Siteswap?
    Basic notation
    Hulling of the trick
    Particular cases : transfer, held ball and empty hand
Properties of the siteswap
Validation of a sequence siteswap
How to check if a sequence is valid?
Siteswap and 3 balls
Siteswap and multiplex tricks
Synchronous Siteswap and tricks
Showers and cascades: the same family

Some examples of tricks in siteswap

Short history

It is in 1985 that 3 American and 1 English (Bruce Tiemann (*), Bengt Magnusson, Paul Klimek, and Mike Day) impassioned juggling and/or mathematics had the idea to create a language specific to this discipline which was to make it possible to codify any trick of juggling simply. Thanks to Internet and to its forums, they effectively could work out their theory of the siteswap; you can find besides in the archives of the web site, some their discussions on the subject.

(*) Bruce Tiemann "Boppo" of the university of Collorado perform an astonishing flash with 11 balls

Why siteswap?

Traditional description consisting in explaining by the text or the word is obligatorily heavy and can be differently interpreted according to the individual; indeed juggling being mainly visual, it is not obvious, even by adding drawings, to describe precisely and simply each movement (sometimes fast) constituting a trick; whereas the interpretation of a trick described in siteswap is easy and nonambiguous, there are even automats (PC software for example) which make it possible to visualize any trick described in siteswap.

Since, many mathematicians or data processing specialists who worked around these magic numbers, many forums on Internet cover subject, some impassioned work on this topic by associating with the siteswap the Topology and the Groups' Theory. In particular Edward Carstens (University of Missouri) which developed a wide notation of SiteSwap called MHN (Multiple Hand Notation) containing matrices with 3 dimensions which makes it possible to codify tricks in "passing" (several jugglers at the same time). It is moreover the author of a powerful data-processing generator of tricks in MHN (JP2).

The artists of modern circus start to integrate in their shows, new tricks deduced from SiteSwap; in particular the tricks in "passing" (2,3 even 4 jugglers at the same time).


The siteswap describes precisely " which, what, when " i.e. the hand which launches (right or left), that which receives, the number of launched objects, the moment of the throw, and its amplitude or duration of flight (called Airtime).

For the moment this coding does not describe " or, how ", i.e. no description of hand's location (examples: claw, snatch or chop) nor hands' movements (examples: crossed hands or behind the back); and it is there the only defect of Siteswap ; but which is serious because certain very visual tricks are only the result of a particular movement of the hands (for example Mill' S Mess). In spite of that, SiteSwap offers many technical possibilities which without it, would never have been suspected.

Basic notation

All is based on the description of the couple (space, time):

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)
we call " 1 Time " the deadline between 2 throws (D G or G D), the combination of juggling is like the music composing, the siteswap is its partition

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)

The trick is not proportional to the height, but to the duration, thus a throw of 6
will last twice longer than a throw of 3; the height will be more of the double

Moreover, this trick being expressed in unit of time (function thus of the rate/rhythm imposed by the juggler), the same trick carried out more quickly (on a short tempo) or more slowly (on a long tempo) will have the same Siteswap notation.


The sequence quoted in example above can be simplified:

This only number 3 is enough to describe the 3 balls cascade, whereas we would have needed a big number of lines of explanation, to describe it by the text.

In the same way sequence 505055050550505 ... is simplified into 50505 or 05055 or 50550 or 55050, by principle we will retain the last because we arranges the sequence in the order of the decreasing digit (without reversing digit). Another example, among these three equivalent forms 423, 234 or 342, the " standardized " notation is the first.

Hulling of the trick

Let us take the example of the 3 balls cascade, it's noted by : 3

While developing we obtain: D3 G3 D3 G3 D3 G3 ...

we can easily represent the way of balls:

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)

we can noted that any launched ball changes hand (normal because the duration of flight is odd, hand inevitably is changed), any ball is launched every 3 times, and their trajectory crosses.

Particular cases: transfer, kept ball and empty hand

With this notation it misses some particular elements which appear in various tricks, a such empty hand, a ball transferred immediately or a ball which is not launched.

These particular cases are to be regarded as real throws:

- empty hand : this throw is noted 0

- transfer or ball which changes hand immediately : this throw is noted 1

- not launched or kept ball : this throw is noted 2

The durations of flight expressed here for these particular cases correspond well to reality. For example, if the juggler keeps a ball in hand instead of the throw, it will wait exactly 2 times before being able to start again it.

Thanks to these particular cases, all tricks can be coded even if they do not utilize only one hand. For example 2 balls with 1 hand is noted40 (either D4 G0 D4 G0 D4 G0......). We will note here that the duration of flight of 4 indicates that the ball falls down in the same hand (this is true for any throw of an even value). Moreover the choice is left to the juggler choose which hand will launch (we could interpret 40 by G4 D0 G4 D0 G4 D0....)

Properties of the siteswap


Validation of a siteswap sequence

A Siteswap sequence is known as valid if it can be juggled, not with the meaning "Is there a juggler who can perform it ? " but with the meaning "its execution is physically possible".

The only postulate which makes a Siteswap sequence valid is:

The various throws described in the sequence should not cause collision between objects. There is collision if at least 2 objects arrive at the same moment in the same hand.

Foot-note: the siteswap is theoretical, the size of the objects does not change the reliability of siteswap, therefore in practice we can see collisions even if the sequence is valid. For example : it is humanly possible to juggle with 11 balls, whereas with 11 clubs it is impossible (try to juggle with 3 balloons in cascade, it is much more difficult than with 3 balls).

Example of nonvalid sequence: 432

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)

It is clear that the first "3" throws lead together on the same hand, which is the sign of triple collision, the correct sequence is 423

Here the graph of the correct sequence 423:

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)

Here, no collision is visible.

we can note that only one ball is swapped between Right hand and Left hand (here ball 3), the other balls (1 and 2) do not change a hand.

How to check if a sequence is valid?

This is the sequence siteswap to be validated: B1 B2 B3 B4.....BN.

This sequence has one period N (it is in fact the length of the sequence after simplification)

We add at each element its row in the sequence, in fact its throw's time, thus we add 0 to B1, 1 in B2... (n-1) with Bn.

A new sequence then is obtained: C1 C2 C3 ... Cn

We simplifie each term C1 C2... Cn by the modulo of N (while Ci is >= N, we subtract N)

The result obtained is a new sequence: D1 D2 D3... Dn

If D1 D2 D3... Dn is a permutation of 0 1 2... N, the sequence is valid. Indeed if there is 2 identical Di elements it is the sign of a collision.

Nothing understood? let us take again the preceding example :

For 432: 432 + 012 = 444  is 444 (modulos 3) = 111 à there is a triple collision

But for 423: 423 + 012 = 435  is 435 (modulo 3) = 102 à no collision, the sequence is valid

Important remark :En practical, before applying the preceding demonstration, we check initially the property which wants that the average of the tricks composing the sequence is not fractionnal; indeed the number of objects used is obligatorily an integer value, for example 532 is not valid beacuser its average is 3,33.

Siteswap and 3 balls

There are often differences between the theory and reality, the siteswap does not escape from this rule, and in particular with 3 balls tricks.

Small explanation, a normally juggler has 2 hands, with using 3 balls,he is often in the position where he has 2 balls in hands and the third is in the air, this ball can have advance or delay in the trick, that will not have any consequence on its good réalization. To the contrary this delay can bring an even spectacular unexpected result; moreover much of very visual tricks with three balls are based on changes of rate/rhythm in the throws.

It is one of the reasons which make that the same trick performedt by different jugglers has not often the same effect. This is why the most beautiful visual effect are performed with only 3 balls and rarely more. The juggler can express himself fully while exploiting change of rates/rhythms, unexpected positions of hands which make him a true illusionnist.

This is much less true starting from 4 balls; because a little desynchronism in the trick leads to its failure, the juggler must respect the rate/rhythm of the throws such as they are described by Siteswap.


The trick 423 which authorizes a lot of combinations according to the tuning of the arms, is very influenced also by the duration of the 2; by respecting the siteswap, we obtain the equivalent trick but slower with the sequence 42522 but there is an infinity of other situations between 423 and 42522 that unfortunately the siteswapp does not enable us to write (for example: 4 2,5 4,5 2 2). For example Fake alternate which is containing 423 but with a throw "3" which is have the same value of a "4" in duration (so that the trick is beautiful, it is necessary that all the throws are identical)

Another example "Rubenstein's Revenge" 52233 (see paragraph : Examples of tricks with Siteswap) it is possible to make a shorter throw 5, to decrease the duration of the 2 and to obtain a more "grouped" trick. The trick called "Tennis" uses this same Siteswap 52233, and here it is even more obvious, the juggler can reduce the duration of "5", just a little more of the "3", so that the ball " tennis " passes just to the top of the 2 others balls; in this case we could note without respecting the siteswap 3,2 2,9 2,9 . Indeed we notes this trick 3.

In conclusion, we can say that for the tricks with 3 balls the siteswap is too constraining on the regular rate/rhythm imposed on the throws, the juggler knows that by modifying the rate/rhythm slightly it will be able " to ideally smooth " his trick and to make it more visual; the viewer softwares of Siteswap do not allow this "tuning", because they interpret nominally and precisely the sequence to be juggled.

On the other hand, starting from 5 balls or more and/or 4 hands (passing 2 jugglers) or more, it is obligatory to respect the good rate/rhythm to succeed in " holding " the trick, indeed the least shift in the timing is very difficult to recover, for a light delay or advance in a throw will influence the following ball.

Siteswap and multiplex tricks

Up to now the siteswap enabled us to codify tricks simplex, for which there is to the more 1 ball by launching; an interesting alternative of jugglings makes it possible to launch 2, 3 even 4 balls at the same time; the constraint of " not collision " presented previously must be respected, for this the balls launched at the same have different AirTime.

The principle is as follows:



-1- Let us start with a simple trick, on the rate/rhythm of the 3 balls cascade but with 5 balls in hands: [32]
Each throw uses 2 balls, 1 ball ("3") is launched in the other hand, the other ball ("2") is preserved in hand.

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)

-2- the trick multiplexing with 5 balls [54][22]2 is one of most traditional; we launche 2 balls each time, one of the 2 balls is crossed, the other returns in the same hand (so on)

(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)

It is less obvious to describe, we can notice: any ball that is launched (the 5 or 4), is then preserved in hand (the same hand for the 4, the other hand for the 5) and that all the balls turn and mix.

Synchronous Siteswap and tricks

Some tricks in juggling require to perform simultaneous throws (right hand and left hand launch at the same time), basic Siteswap notation does not make it possible to codify this type of tricks, indeed it obliges to perform the throws asynchronous (right hand and after left hand and so on .........).

The principle of the SiteSwap notation of the synchronous throws is as follows.


-1- the 2 balls trick which consist to swap simultaneously the balls in each hand is noted:


The time scale is divided by 2 because there are only even throws.

-2- Another example, the box with 3 balls is noted:

(4,2X) (2X,4)

we can notice that only one ball changes hand, it is the ball 2, it is launched twice more often as the 2 others (indeed its duration of flight being 2 times less, to compensate it should be launched twice more often)

-3- Last example, the half-shower 3 balls: it is like a traditional shower except that the throws are simultaneous (this trick gives the impression of a wheel without end):


The preceding example (the box) is in fact the association of the beginning of 2 half-showers but in the opposite directions.

Showers and cascades: the same family

As odd as that can appear, the showers and cascades belong in fact to the same great family of tricks, that called wrongly "half-showers" (helf-shower, half-cascade).

To understand, let us develop the siteswap sequences for 3, 4 and 5 balls (with just the combinations length to 2).

With 3 balls we obtain 4 different tricks :    33 (traditional cascade with 3 balls)
42 noted (4x,2x)    (half-shower with 3)
51           (shower with 3)
60           (shower in 1 hand)

With 4 balls we obtain 5 different tricks :    44 noted (4x,4x)  (cascade)
53 (half-shower)
62 noted (6x,2x)  (half-shower)
71            (shower)
80            (shower)

With 5 balls we obtain 6 different tricks :    55       (cascade)
64 noted (6x,4x)   (half-shower)
73 (half-shower)
82 noted (8x,2x)   (half-shower)
91        (shower)
A0       (shower)
so on for 6, 7 balls and more...


The basic shower with 3 balls notes 51,  it is a question of replacing the ball from the " 1 " by a throw a little higher " 2x ", and the " 5 " by a throw a little low " 4x ", then we obtain the half shower which is noted (4x,2x). If we applie this rule a second time, the " 4x " becomes 3 and the " 2x " also, then we obtain 33 or 3 (3 balls cascade). We have passed from the shower to the cascade (or conversely) just by modulating the amplitudes of the throws: we can conclude from it that we are in the same family of tricks.

Identical explanations for 4 and 5 balls

Just an interresting detail for 4 balls: we does not obtain the fountain " 4 " but well a cascade noted in this case (4x,4x) caution ! with the possible collision (which said that the cascade with 4 did not exist ? ); this is true for all the even numbers of ball.

Siteswap and tricks in passing

(to be come)

Theorem of Shannon (F+D)H=(V+D)N

Siteswap and balls with rebound

(to be come)

Transitions between tricks

(to be come)

Another notations

(to be come)

Generators and interpreters

(to be come)


(to be come)

Some examples of tricks in siteswap

- 1 ball

the examples with 1 ball which follow are interesting just for the teaching aspect of the siteswap

-1 : the ball passes quickly between one hand to the other

-20 : a hand which keeps the single ball (without throw), the other hand is empty

-4000 : the ball is always launched by the same hand

-600000 : the same but higher

-300 : the ball is launched and changes hand

-50000 : the same but higher

- 2 balls

the 6 examples with 2 balls which follow are interesting just for the teaching aspect of the siteswap

-22 : a ball in each hand without launching

- [22]0 : 2 balls a hand without launching

- [42]020 : 2 balls a hand, 1 threw by the same hand, the other emptied

-4202 : a ball stopped in a hand, the other always threw by the same hand

- [64]00020 : 2 balls in a hand, threw simultaneously by the same hand

- [ 53]0020 : idem but by changing hand

the following sequences are interesting to juggle

-40 : 2 balls a hand

-8000 : idem but higher

-330 or 303 : 2 balls snake (3 balls cascades with a hole)

-31 : 2 balls shower

- (2X,2X) : 2 balls half-shower (simultaneous exchange 1 ball in each hand)

- (4,2X)(2X,0)(2X,4)(0,2X) : box like

- (8,2X)(4,0)(0,0)(2X,0)(2X,8)(0,4)(0,0)(0,2X) : 2 balls extended box


- 3 balls

-3 : traditional 3 balls cascade (but also reversed Cascade, Mills' Mess, Boston Mess etc)

-522 : slow cascade

-72222 : very slow cascade

-55500 : 3 balls flash (start of a 5 balls cascade)

-55050 or 50505 : 3 balls snake (started from a 5 balls cascade)

-42 : 2 balls in a hand, 3rd ball kept without launching in the other hand. This apparently simple trick authorizes a lot of fantasy by moving the hand which keep the ball (YO-YO, OY-OY, Fake etc...)

-(4,4)(4,0) : pistons, 3 balls in column with 2 balls in a hand and 1 ball in the other (this trick authorize a lot of possibilities according to the type of launch)

- (4,4)(4X,0)(4,4)(0,4X) : idem but with change of hand

- (6,6)(2X,0)(0,4)(6,6)(0,2X)(4,0) : idem but the ball passes quickly in the other hand, and is launched external

-423 : (Rigth-Middle-Left) this simple trick brings spectacular effects by moving the hand which keep the ball "2" (Burkes Barrage, Wave, alternated Fake, Piano, Follow, etc....)

-441 : an interesting trick which remembered 4 balls trick, the effects are different according the throws "4" (fountain type, outside or interior or alternate)

-612 : asynchronous box, approaches the traditional box with nonsimultaneous throws

- (4,2X)(2X,4) : traditional box

- (4X,2X)(4,2X)(2X,4X)(2X,4) : double box (attention, to perform 1/8 of turn or to move your arms between each box

-51 : 3 balls shower

- (4X,2X) : 3 balls half-shower and siteswap of Shuffle (ball 2x is launched between 2 balls 4x)

-4440 : 4 balls fountain with a hole

-60 : 3 balls in a hand

-52233 : siteswapp of tennis (cascades to 3 with a ball, always the same one, which makes outside) and siteswap of Rubenstein's Revenge (derivative of Mills' Mess with very specific position of the hands)

- 4 balls

(to be come)

- 5 balls

(to be come)

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