# Siteswap

Siteswap (also called Quantum Juggling by the inventor, mathematician Paul Klimek, and Cambridge notation in the United Kingdom) is a juggling notation used to describe or represent juggling patterns. It encodes the number of beats of each throw, which is related to their height, and the hand to which the throw is to be made: "The idea behind siteswap is to keep track of the order that balls are thrown and caught, and _only_ that."[1] It is an invaluable tool in determining which combinations of throws yield valid juggling patterns for a given number of objects, and has led to previously unknown patterns (such as 441). However, it does not describe body movements such as behind-the-back and under-the-leg.

Siteswap beats shown as relative height[2]

The numbers are as follows:[3]

• 0 = "missing"/rest [empty hand]
• 1 = pass [between hands]
• 2 = hold [one hand/no toss]
• 3 = (3-ball) cascade toss [between hands]
• 4 = (4-ball) fountain toss [up and into same hand]
• 5 = high toss [between hands]

For example, a three-ball cascade may be notated "3", while a shower may be notated "5 1".[4] The name siteswap comes from the ability to generate patterns from "swapping" numbers in preexisting patterns, such as 55500 and 50505 (or Flash and Snake).

## Origin

The notation was invented by Paul Klimek in Santa Cruz, California in 1981, and later developed by undergraduates Bruce "Boppo" Tiemann and the late Bengt Magnusson at the California Institute of Technology in 1985, and by Mike Day, mathematician Colin Wright, and mathematician Adam Chalcraft in Cambridge, England in 1985 (whence comes the alternative name).[5][a] The numbers derive from the number of balls used in the most common juggling patterns.

## Vanilla

Its simplest form, sometimes called vanilla siteswap, describes only patterns whose throws alternate hands and in which one ball is thrown at a time. If we were to watch someone from above as they were juggling while walking forward, we might see something like the diagram to the right, which is sometimes called a space-time diagram or ladder diagram. In this diagram, time progresses from the top to the bottom.

We can describe this pattern by stating how many throws later the ball is thrown again. For instance, on the first throw in the diagram, the purple ball is thrown in the air by the right hand, next the blue ball, the green ball, the green ball again, and the blue ball again and then finally the purple ball is caught and thrown by the left hand on the fifth throw, this gives the first throw a count of 5. We end up with a sequence of numbers which denote the height of each throw to be made. Since hands alternate, odd-numbered throws send the ball to the other hand, while even-numbered throws send the ball to the same hand. A 3 represents a throw to the opposite hand at the height of the basic three-cascade; a 4 represents a throw to the same hand at the height of the four-fountain, and so on.

There are three special throws: a 0 is a pause with an empty hand, a 1 is a quick pass straight across to the other hand, and a 2 is a momentary hold of an object. Throws longer than 9 beats are given letters starting with a. The number of beats a ball is in the air usually corresponds to how high it was thrown, so many people refer to the numbers as heights, but this is not technically correct; all that matters is the number of beats in the air, not how high it is thrown. For example, bouncing a ball takes longer than a throw in the air to the same height, and so can be a higher siteswap value while being a lower throw.

Each pattern repeats after a certain number of throws, called the period of the pattern. The pattern is named after the shortest repeating segment of the sequence, so the pattern diagrammed on the right is 53145305520 and has a period of 11. If the period is an odd number, like this one, then each time you repeat the sequence you're starting with the other hand, and the pattern is said to be symmetrical because each hand is doing the same thing (although at different times). If the period is an even number then on every repeat of the pattern, each hand does the same thing it did last time and the pattern is possibly asymmetrical because each hand might be doing something different.

The number of balls used for the pattern is the average of the throw numbers in the pattern.[8] For example 441 is a three-object pattern because (4+4+1)/3 is 3, and 86 is a seven-object pattern. All patterns must therefore have a siteswap sequence that averages to an integer. Not all such sequences describe patterns; for example 543 with integer average 4 but its three throws all land at the same time, colliding.

Some hold to a convention in that a siteswap is written with its highest numbers first. One drawback to doing so is evident in the pattern 51414, a 3-ball pattern which cannot be inserted into the middle of a string of 3-throws, unlike its rotation 45141 which can.

### State diagrams

Just after throwing a ball (or club or other juggling object), all balls are in the air and are only under the influence of gravity. Assuming you catch the balls at a consistent level, then the timing of when the balls land is already determined. We can mark each point in time when a ball is going to land with an x, and each point in time when there is not yet a ball scheduled to land with a -. This describes the state that you're in and determines what you can throw next. For instance, we can look at the state just after our first throw in the diagram, it is xx--x. We can use the state to determine what can be thrown next. First we take the x off the left hand side (that's the ball that's landing next) and shift everything else to the left filling in a - on the right. This leaves us with x--x-. Since we caught a ball (the x we removed from the left) we can't "throw" a 0 next. We also can't throw a 1 or a 4, because there are already balls scheduled to land there. So assuming that the highest we can accurately throw a ball is to a height of 5, then we can only throw a 2, 3, or a 5. In this diagram, the juggler threw a 3, so an x goes in the third spot, replacing the -, and we have x-xx- as the new state.

The diagram on the left shows all possible states for someone juggling three items and a maximum height of 5. From each state you can follow the arrows and the corresponding numbers give you the siteswap. Any path which brings you back to the same state that you started with is a valid siteswap, and all siteswaps can be generated this way. The diagram becomes a lot bigger very fast when more balls or higher throws are introduced as there are more possible states and more possible throws.

Another method of representing siteswap states is having the next throw on the right and read to the left (instead of the left to right system described above), represent a ball with a 1 instead of an x, and represent a spot where there's no ball scheduled to land with a 0 instead of a -. Then the state can be represented with a binary number, such as binary 10011 for the first state in the space-time diagram above. This method makes it a lot easier to represent more objects or higher throws as it is less complicated to create.

 (111) 7 (1011) 11 (1101) 13 (1110) 14 (10011) 19 (10101) 21 (10110) 22 (11001) 25 (11010) 26 7 11 13 14 19 21 22 25 26 28 3 4 5 2 4 5 1 4 5 0 2 3 5 1 3 5 0 1 2 5 0 0

The table on the right contains the same information as the diagram on the left. To generate a siteswap, first find the row of your starting state. Reading across the row will give you the possible numbers you can throw and the column will give you the resulting state. Moving like this from state to state until you end up back in your starting state will give you a valid siteswap.

### Mathematical connections

Vanilla siteswap patterns may be viewed as certain elements of the affine Weyl group of type ${\tilde{A}}_n$. One presentation of this group is as the set of bijective functions f on the integers such that, for a fixed n: f(i + n) = f(i) + n for all integers i. If the element f satisfies the further condition that f(i) ≥ i for all i, then f corresponds to the (infinitely repeated) siteswap pattern whose ith number is f(i) − i: that is, the ball thrown at time i will land at time f(i).

A subset of these siteswap patterns naturally label strata in the positroid stratification of the Grassmannian.[9]

## Extensions

### Multi-handed

Multi-hand notation was developed by Ed Carstens in 1992 for use with his juggling program JugglePro.[6] Site-swap notation in its simplest form ("Vanilla siteswap") assumes that only one ball is thrown at a time. It follows that any valid site-swap for two hands will also be valid for any number of hands, on the condition that the hands throw after each other. Commonly used multi-hand siteswaps are 1-handed (diabolo) site-swap, and 4-handed (passing) site-swap.

1-handed (diabolo)

The site-swap is performed by a single hand, or a diabolo player throwing diabolos at different heights.

4-handed

Valid site-swaps can be juggled by a 4-handed juggler, or for 2 jugglers coordinating 4 hands, on the condition that hands throw alternately.

In practice, this is most easily obtained if the jugglers throw by turns, one sequence being (Right hand of juggler A, right hand of juggler B, left hand of A, left hand of B).

### Synchronous

Siteswap notation can be extended to denote patterns containing synchronous throws from both hands. The numbers for the two throws are combined in parentheses and separated by a comma. Only even numbers are allowed (each throw involves two hands, and thus only even numbers), so throws that move to the other hand are marked by an x. Thus a synchronous three-prop shower is denoted (4x,2x), meaning one hand continually throws a low throw or 'zip' to the opposite hand, while the other continually makes a higher throw to the first. Sequences of bracketed pairs are written without delimiting markers.

### Multiplexing

A further extension allows siteswap to notate patterns involving multiple throws from either or both hands at the same time in a multiplex pattern. The numbers for multiple throws from a single hand are written together inside square brackets. For example, [33]33 is a normal 3-ball cascade, with a pair of balls always thrown together.

### Passing

Simultaneous juggling: <xxx|yyy> notation means one juggler does 'xxx' while another does 'yyy'. 'p' is used to represent a passing throw. For example, <3p 3|3p 3> is a 6 prop '2 count' passing pattern, where all left hand throws are passes and right hand throws are selves. This can also be used with synchronous patterns; a two person 'shower' is then <(4xp,2x)|(4xp,2x)>

If the pattern contains fractions, e.g. <4.5 3 3 | 3 4 3.5> the juggler after the bar is supposed to be half a count later, and all fractions are passes. If both juggle the same pattern (although shifted in time), the pattern is called a social siteswap and only half of the pattern needs to be written: <4p 3| 3 4p> becomes 4p 3 and <4.5 3 3| 3 4.5 3> becomes 4.5 3 3.

### Other extensions

Various other enhancements to the siteswap system have been proposed and enjoy some use.

• Synchronous siteswaps that switch sides: patterns that repeat in mirror image can be abbreviated with a *. For example, Instead of (4,2x)(2x,4) (3-ball box pattern), you can write (4,2x)*.
• In software, files describing patterns may contain additional information. In Joepass, each throw can be given an individual delay, thus allowing the description of irregular rhythms.

## Programs

There are many free computer programs available which simulate juggling patterns.

## Notes

1. ^
• "Invented independently around 1985 by Paul Klimek of the University of California at Santa Cruz, Bruce Tiemann of the California Institute of Technology and Michael Day of the University of Cambridge."[4]
• "Invented around 1985 by three people independently: Bruce "Boppo" Tiemann at Caltech, Paul Klimek in Santa Cruz, and Mike Day in Cambridge."[1]
• "...Bruce Tiemann (Boppo) and the late Bengt Magnusson....Other contributors to the development of site swap theory include Jack Boyce, Allen Knutson, Ed Carstens, and jugglers on the computer network."[6]
• "Jack Boyce (also at Caltech) came up with the juggling state model to explain the phenomenon of excited-state tricks."[1]
• "To give credit where it is due, the notation as presented here was independently (and previously) invented by Paul Klimek, with whom we have had helpful discussions."[7]

## References

1. ^ a b c Knutson, Allen. "Siteswap FAQ", Juggling.org.
2. ^ After Donahue, Bill (2004). "The Mathematics of...Juggling", DiscoverMagazine.com.
3. ^ "Juggling by numbers: How notation revealed new tricks", BBC.com (BBC article with explanatory video)
4. ^ a b Beek, Peter J. and Lewbel, Arthur (1995). "The Mathematics of Juggling", part of "The Science of Juggling", Scientific American. Also at Juggling.org.
5. ^ Hayes, David F.; Shubin, Tatiana (2004). Mathematical Adventures for Students and Amateurs. The Mathematical Association of America. p. 99. ISBN 0-88385-548-8. Retrieved February 17, 2011.
6. ^ a b Lewbel, Arthur (1996). "The Academic Juggler: The Invention Of Juggling Notations", Juggle.org.
7. ^ Tiemann, Bruce and Magnusson, Bengt (1991). "A Notation for Juggling Tricks. A LOT of Juggling Tricks.", Juggle.org.
8. ^ Tiemann, Bruce and Magnusson, Bengt (1991). "A Notation for Juggling Tricks, A LOT of Juggling Tricks", Juggle.org. Accessed July 8, 2014.
9. ^ Knutson, Allen; Lam, Thomas; Speyer, David. "Positroid Varieties: Juggling and Geometry". arXiv:1111.3660.