Summary:
Short historyShort history
Why Siteswap?
Basis
Basic notation
Simplification
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 familySome examples of tricks in siteswap
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
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.
All is based on the description of the couple (space, time):
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
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.
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:
- transfer or ball which changes hand immediately : this throw is noted 1
- not launched or kept ball : this throw is noted 2
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....)
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
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
(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)
Here the graph of the correct sequence 423:
Here, no collision is visible.
(Sorry, for the french legend on the graphic: Temps = Time, D (Droite) = Right, G (Gauche) = Left)
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.
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.
Examples:
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.
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:
Examples:
-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:
(2X,2X)
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):
(4X,2X)
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...
Explanations:
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)
(to be come)
(to be come)
(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- 2 balls-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
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
- 4 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, 3^{rd} 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)
(to be come)
- 5 balls
(to be come)