Siteswap

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".

Origin The notation was invented by Paul Klimek in Santa Cruz, California in 1981, and later developed by undergraduates Bruce "Boppo" Tiemann, Joel David Hamkins, and the late Bengt Magnusson at the California Institute of Technology in 1985, and by Mike Day, mathematician Colin D Wright, and mathematician Adam Chalcraft in Cambridge, England in 1985 (whence comes an alternative name). Hamkins wrote computer code in 1985 to systematically generate siteswap patterns—the printouts were taken immediately to the Athenaeum lawn at Caltech to be tried out by himself, Tiemann, and Magnusson. The numbers derive from the number of balls used in the most common juggling patterns. Siteswap has been described as, "perhaps the most popular" name. The name siteswap comes from the ability to generate patterns by "swapping" landing times of any 2 "sites" in a siteswap using the swap property. For example, swapping the landing times of throws "5" and "1" in the siteswap "51" generates the siteswap "24".

Vanilla

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.

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. 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.

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. Since synchronous throws are only thrown on even beats, only even numbers are allowed. 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)*.

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

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