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Notation used to describe juggling patterns

Siteswap, also called quantum juggling or the Cambridge notation, is a numeric juggling notation used to describe or represent juggling patterns. The term may also be used to describe siteswap patterns, possible patterns transcribed using siteswap. Throws are represented by non negative integers that specify the number of beats in the future when the object is thrown again: "The idea behind siteswap is to keep track of the order that balls are thrown and caught, and only that." 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 assumes that "throws happen on beats that are equally spaced in time." For example, a three ball cascade may be notated "3 ", while a shower may be notated "5 1". efn|
"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." "Invented around 1985 by three people independently: Bruce "Boppo" Tiemann at Caltech, Paul Klimek in Santa Cruz, and Mike Day in Cambridge." "Jack Boyce (also at Caltech) came up with the juggling state model to explain the phenomenon of excited state tricks." The name siteswap comes from the ability to generate patterns by "swapping" landing times of any 2 "sites" in a siteswap using the For example, swapping the landing times of throws "5" and "1" in the siteswap "51" generates the siteswap "24".

Its simplest form, sometimes called vanilla siteswap, describes only patterns whose throws alternate hands and in which one ball is thrown from each hand at a time. If one were juggling while walking forward, something like the adjacent diagram would be seen from above, sometimes called a space time diagram or ladder diagram. In this diagram, three balls are being juggled. Time progresses from the top to the bottom. This pattern can be described by stating how many throws later each ball is caught. For instance, on the first throw in the diagram, the purple ball is thrown in the air (up out of the screen, towards the bottom left) 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. This produces 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. +Siteswap Throw NamesThrow NameBeats object is in AirSwitches handsDescription0 Empty hand11YesThrow from one hand to the other20NoMomentary hold33YesThrow from a 3 ball cascade44NoThrow from a 4 ball fountain55YesThrow from a 5 ball cascade66NoThrow from a 6 ball fountain77YesThrow from a 7 ball cascade88NoThrow from a 8 ball fountain99YesThrow from a 9 ball cascadea10NoThrow from a 10 ball fountainb11YesThrow from a 11 ball cascade
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 period is the number of digits in the shortest non repeating representation of a pattern. For example, the pattern diagrammed on the right is 53145305520 which has 11 digits and therefore has a period of 11. If the period is an odd number, like this one, then each time the sequence is repeated, the sequence starts with the other hand, and the pattern is 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 asymmetrical. The number of balls used for the pattern is the average of the throw numbers in the pattern. Throws that move to the other hand are marked by an x following the number. 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. Patterns that repeat in mirror image on the opposite side can be abbreviated with a *. For example, Instead of (4,2x)(2x,4) (3 ball box pattern), can be abbreviated to (4,2x)*.

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.

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)>

New valid vanilla sequences can be generated by swapping adjacent elements from another valid vanilla siteswap sequence, adding 1 to the number being swapped to the right and subtracting 1 from the number being swapped to the left. The longest prime patterns with height 22 (with 3 ball maximum), for 9 balls (with 13 maximum height), and for heights and ball counts in between, were enumerated by Jack Boyce in February 1999 using a program called jdeep. The full list of longest prime siteswaps generated by jdeep (with '0' throws represented by a ' ' and maximum height throws represented by a '+') can be found here.

Vanilla siteswap patterns may be viewed as certain elements of the affine symmetric group (the affine Weyl group of type ). 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.

Number: Relative duration (height) of a toss. 1, 2, 3
Brackets []: Multiplex. [333]33. Chevrons and vertical bar <|>: Simultaneous and passing patterns. P: Pass. <333P|333P>
Fraction: Pass 1/y beats later. <4.5 3 3 | 3 4 3.5>
Parentheses : Synchronous pattern. *: Synchronous pattern that switches sides. (4,2x)(2x,4) = (4,2x)*
x: Toss to the other hand during a synchronous pattern.