Siteswap: Difference between revisions

'''Siteswap''', also called '''quantum juggling''' or the '''Cambridge notation''', is a numeric [[juggling notation]] used to describe or represent [[juggling pattern]]s. 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."<ref name="FAQ">{{Cite web|url=http://www.juggling.org/help/siteswap/faq.html|title=Siteswap FAQ|last=Knutson|first=Allen|website=[[Juggling Information Service|Juggling.org]]|access-date=June 30, 2017}}</ref> 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 [[beat (music)|beats]] that are equally spaced in time."<ref name="B&L">{{Cite bookjournal|last1=Beek|first1=Peter J.|last2=Lewbel|first2=Arthur|chapter=The Mathematics of Juggling|title=The Science of Juggling|chapter-url=https://www2.bc.edu/~lewbel/jugweb/sciamjug.pdf|archive-url=https://web.archive.org/web/20160304104003/https://www2.bc.edu/~lewbel/jugweb/sciamjug.pdf|archive-date=March 4, 2016|url-status=dead|journal=[[Scientific American]]|issn=0036-8733|language=en|date=November 1995|volume=273|issue=5|pages=92–97|doi=10.1038/scientificamerican1195-92|bibcode=1995SciAm.273e..92B|jstor=24982089}} Also available at [http://www.juggling.org/papers/science-1/mathematics.html Juggling.org].</ref>

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.