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.

"[3] 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).

Siteswap assumes that "throws happen on beats that are equally spaced in time.

[4] 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).

[5][a] 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.

This pattern can be described by stating how many throws later each ball is caught.

This produces a sequence of numbers which denote the height of each throw to be made.

The period is the number of digits in the shortest non-repeating representation of a pattern.

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).

Not all such sequences describe patterns; for example 543 with integer average 4 but its three throws all land at the same time, colliding.

Siteswap notation can be extended to denote patterns containing synchronous throws from both hands.

[10] Throws that move to the other hand are marked by an x following the number.

Patterns that repeat in mirror image on the opposite side can be abbreviated with a *.

The numbers for multiple throws from a single hand are written together inside square brackets.

Multi-hand notation was developed by Ed Carstens in 1992 for use with his juggling program JugglePro.

This describes the current state and determines what number ball can be thrown next.

Since we caught a ball (the x we removed from the left) we can't "throw" a 0 next.

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 shown illustrates all possible states for someone juggling three items and a maximum height of 5.

From each state one can follow the arrows and the corresponding numbers produce the siteswap.

A siteswap state diagram can also be represented as a state-transition table, as shown on the right.

This process can be repeated so that when the original state is reached, a valid siteswap will be created.

Since the transitions in this sequence create a cycle in the graph, this pattern is valid.

There are other methods of determining a sequence's validity based on the flavor of siteswap.

To find if a pattern is valid, first create a new sequence formed by adding

A non-rigorous but simpler method of determining if a siteswap is prime is to try to split it into any valid shorter pattern which uses the same number of props.

Sometimes this process does not work; for example, 153 (better known by its rotation 531) looks like it can be split into 15 and 3, but checking that the cycle has no repeating nodes in the graph traversal indicates that it is prime by the more rigorous definition.

It has been shown empirically that the longest prime siteswaps bounded by height

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.