Siteswap: Difference between revisions

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 Wright, and mathematician Adam Chalcraft in [[Cambridge|Cambridge, England]] in 1985 (whence comes an alternative name).<ref>{{Cite book|url=https://books.google.com/books?id=PD0clAlF8O4C&pg=PA99|title=Mathematical Adventures for Students and Amateurs|date=2004|publisher=Mathematical Association of America|last1=Hayes|first1=David F.|last2=Shubin|first2=Tatiana|isbn=0883855488|oclc=56020214|page=99}}</ref>{{efn|

*"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."<ref name="T&M"/>}} Hamkins wrote computer code in 1985 to systematically generate siteswap patterns&mdash;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.<ref name="Sethares">{{Cite book|title=Rhythm and Transforms|url=https://archive.org/details/rhythmtransforms00seth_415|url-access=limited|last=Sethares|first=William Arthur|author-link=William Sethares|date=2007|publisher=[[Springer Publishing|Springer]]|isbn=9781846286407|oclc=261225487|page=[https://archive.org/details/rhythmtransforms00seth_415/page/n52 40]}}</ref>

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 we were to watch someone from above as theyone were juggling while walking forward, we might see something like the adjacent diagram, whichwould isbe 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.

Each pattern repeats after a certain number of throws, called the ''[[Frequency|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 youthe repeatsequence is repeated, the sequence you're startingstarts 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'''.

===Multi-handed===

===Multiplexing=State diagrams==

Not all siteswap sequences are valid.<ref name=":1" /> All Vanillavanilla, synchronous, and multiplex siteswapssiteswap sequences are valid if their state transitions create a cycle in their state diagram graph.<ref name=":1" /> Sequences that do not create a cycle are invalid. For example, Thethe pattern 531 can be mapped to a state diagram as shown on the right. Since the transitions in this sequence create a cycle in the graph, this pattern is valid.

=== Swap Propertyproperty ===

New valid synchronousvanilla 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.<ref name=":0" /> Without loss of generality, theThe swap property will convert the valid sequence <math display="inline">a_0a_1a_2...a_ia_{i+1}...a_{n-1}</math> with arbitrary value <math>i</math>, to generate the new valid sequence <math>a_0a_1a_2...(a_{i+1}+1)(a_i-1)...a_{n-1}</math>.

=== Slide Propertyproperty ===

A valid synchronous sequence can be converted to a valid asynchronous sequence and vice versa using the slide property. Given the synchronous sequence <math>(a_0,a_1)(a_2,a_3)...(a_{n-2},a_{n-1})</math>, thetwo new vanilla sequences can be formed: <math display="inline">b_0 b_1 ... b_{n-1}</math> whereand <math>c_0 c_1 ... c_{n-1}</math>, where<math display="block">b_i = \begin{cases} a_i+1, & \text{if }i\text{ is even and }a_i\text{ crosses hands} \\ a_i-1, & \text{if }i\text{ is odd and }a_i\text{ crosses hands} \\ a_i, & \text{otherwise} \end{cases}</math>and <math>c_0 c_1 ... c_{n-1}</math> where<math display="block">c_i = \begin{cases} a_{i+1}+1, & \text{if }i\text{ is even and }a_i\text{ crosses hands} \\ a_{i-1}-1, & \text{if }i\text{ is odd and }a_i\text{ crosses hands} \\ a_{i+1}, & \text{if }i\text{ is even and } a_i \text{ does not cross} \\ a_{i-1}, & \text{if }i\text{ is odd and } a_i \text{ does not cross}\end{cases}</math>The slide property gets its name by sliding the throw times of one of the hands by one time unit so that the throws align asynchronously.<ref name=":1" /> For example, the siteswap (8x,4x)(4,4) would create two asynchronous (vanilla) siteswaps using the slide property: 9344 and 5744.

*Number: Relative duration (height) of a toss. 1, 2, 3...

*[httphttps://wwwydgunz.gunswapgithub.coio/gunswap/ Gunswap Juggling] (Online animator)

*[http://www.twjc.cojugglingedge.ukcom/calculator.html TWJC Siteswap Calculator] (Helpful Vanilla, Multiplex and Synchronous siteswap validator)