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[[File:Siteswap relative visualized.png|thumb|Siteswap beats shown as relative height<ref>{{Cite news|last=Donahue|first=Bill|date=December 3, 2004|title=The Mathematics of... Juggling|work=[[Discover Magazine]]|url=http://discovermagazine.com/2004/dec/mathematics-of-juggling|access-date=June 30, 2017}}</ref><ref name="T&M">Tiemann, Bruce and Magnusson, Bengt (1991). "[https://web.archive.org/web/20120612161501/https://www.juggle.org/history/archives/jugmags/43-2/43-2,p31.htm A Notation for Juggling Tricks, A LOT of Juggling Tricks]", ''Juggle.org''. Accessed July 8, 2014. [https://www.juggle.org/history/archives/jugmags/43-2/43-2,p31.htm original url]</ref>]]
'''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 [[
For example, a three-ball [[Cascade (juggling)|cascade]] may be notated "3 ", while a [[shower (juggling)|shower]] may be notated "5 1".<ref name="B&L"/>
==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 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|
*"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."<ref name="B&L"/>
*"...site swap patterns...in the form invented by some of Bruce Tiemann, Bengt Magnusson, and Joel Hamkins"<ref name="IBM">{{Cite web|url=http://www.juggling.org/programs/ibm-pc/|title=(in) IBM-PC Programs|last=Knutson|first=Allen|website=[[Juggling Information Service|Juggling.org]]|access-date=October 3, 2023}}</ref>
*"Invented around 1985 by three people independently: Bruce "Boppo" Tiemann at Caltech, Paul Klimek in Santa Cruz, and Mike Day in Cambridge."<ref name="FAQ"/>
*"...Bruce Tiemann (Boppo) and the late Bengt Magnusson....Other contributors to the development of site swap theory include Jack Boyce, Allen Knutson, Ed Carstens, and jugglers on the computer network."<ref name="Lewbel">Lewbel, Arthur (1996). "[http://www.juggle.org/history/archives/jugmags/49-2/v49no2-page%2057.htm The Academic Juggler: The Invention Of Juggling Notations] {{webarchive|url=https://web.archive.org/web/20140714182339/http://www.juggle.org/history/archives/jugmags/49-2/v49no2-page%2057.htm |date=July 14, 2014 }}", ''Juggle.org''.</ref>
*"Jack Boyce (also at Caltech) came up with the juggling state model to explain the phenomenon of excited-state tricks."<ref name="FAQ"/>
*"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—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>
The name ''siteswap'' comes from the ability to generate patterns by "swapping" landing times of any 2 "sites" in a siteswap using the {{Pslink|swap property|}}.<ref>{{Cite web|last=Boyce|first=Jack|date=October 11, 1997|title=Patterns from Lodi 1997 Workshop|url=http://www.sonic.net/~boyce/lodilist.html|url-status=dead|archive-url=https://web.archive.org/web/20041207193727/http://www.sonic.net/~boyce/lodilist.html|archive-date=December 7, 2004|access-date=2020-07-08|website=sonic.net}}</ref> For example, swapping the landing times of throws "5" and "1" in the siteswap "51" generates the siteswap "24".
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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
{| class="wikitable"
|+Siteswap Throw Names
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[[File:Juggling - 3-ball box (4,2x)(2x,4) ladder diagram.svg|150px|thumb|Ladder diagram for box: (4,2x)(2x,4)|alt=]]
Siteswap notation can be extended to denote patterns containing synchronous throws from both hands. The numbers for the two throws are combined in [[Bracket#Parentheses
==Multiplexing==
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If both juggle the same pattern (although shifted in time), the pattern is called a social siteswap and only half of the pattern needs to be written: ''<4p 3| 3 4p>'' becomes ''4p 3'' and ''<4.5 3 3| 3 4.5 3>'' becomes ''4.5 3 3''. (note that in the latter case, 4.5 will be straight passes from one juggler, crossing passes (i.e. left to left or right to right hand) from the other juggler.
Social siteswaps can also be created for more than 2 jugglers (e.g. 4p 3 3 or 3.7 3 for 3 jugglers, where 3.7 is meant to mean 3.66666.... or 3 {{frac|2|3}}. Then each juggler should start {{frac|1|3}} count after the previous one.)
Note that some jugglers use fractions to note multi-handed patterns.
==Multi-handed==
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===Validity===
[[File:Siteswap 531 state diagram.png|thumb|Siteswap 531 state diagram]]
Not all siteswap sequences are valid.<ref name=":1" /> All vanilla, synchronous, and multiplex siteswap 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,
There are other methods of determining a sequence's validity based on the flavor of siteswap.
A '''vanilla''' siteswap sequence <math>a_0a_1a_2...a_{n-1}</math>where <math>n</math> is the period of the siteswap, is valid when the [[cardinality]] of the set <math>S</math> (written in [[Set-builder notation]]) is equal to the period <math>n</math> where<math display="block">S=\{(a_i+i)\bmod n | 0\leq i\leq n-1\}</math>To find if a pattern is valid, first create a new sequence formed by adding <math>0</math> to the first number, <math>1</math> to the second number, <math>2</math> to the third number and so on. Second, calculate the modulus (remainder) of each number with the period. If none of the numbers are duplicated in this final sequence, then the pattern is valid.<ref name=":0">{{Cite web|url=https://www.qedcat.com/articles/juggling_survey.pdf|title=The Mathematics of Juggling|last=Polster|first=Burkard|date=|website=qedcat.com
For example, the pattern 531 would produce <math>5+0,3+1,1+2 </math> or <math>5,4,3</math>. Since the pattern 531 has a period of 3,
A '''synchronous''' siteswap is valid if
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=== Slide property ===
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>,
=== Prime patterns ===
Siteswaps may be considered either prime or composite.<ref name=":1" /> A siteswap is prime if the path created in its state diagram does not traverse any state more than once. Siteswaps that are not prime are called composite.
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.<ref name=":1" /> For example, 44404413 can be split into 4440, 441, and 3; therefore, 44404413 is a composite. Another example, 441, which uses three props, is prime, as 1, 4, 41, and 44 are not valid three prop patterns (as 1/
It has been shown empirically that the longest prime siteswaps bounded by height <math>h</math> contain mostly the throws <math>0</math> and <math>h</math>.<ref>{{Cite web|url=https://www.jonglage.net/theorie/notation/siteswap-avancee/refs/Jack%20Boyce%20-%20The%20Longest%20Prime%20Siteswap%20Patterns.pdf|title=The Longest Prime Siteswap Patterns|last=Boyce|first=Jack|date=|website=jonglage.net
== Mathematical connections ==
=== Connections to abstract algebra ===
Vanilla siteswap patterns may be viewed as certain elements of the [[affine symmetric group]] (the [[affine Weyl group]] of type <math>{\tilde{A}}_n</math>).<ref>{{Cite journal|
=== Connections to topology ===
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*[http://www.jongl.de/ Jongl] - 3d animator capable of displaying multihand (passing) patterns.
*[http://koelnvention.de/w/?page_id=151 JoePass!] works on Windows, Macintosh and Wine (For Linux)
*[
There are also some games to play with siteswap:
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*[http://www.artofdiabolo.com/dsss/ DSSS: The Diabolo Siteswap Simulator], ''ArtofDiabolo.com''.
*[https://jugglinglab.org/ Juggling Lab] (Downloadable animator)
*[
*[http://
*"[http://www.owenreynolds.net/notation/Symmetric_patterns2.pdf#5983DD.pdf Staggered Symmetric Passing Patterns for 2 jugglers]" by Sean Gandini (social siteswaps)
*{{webarchive |url=https://web.archive.org/web/20030806094612/http://pweb.netcom.com/~hjsmith/Juggler.html |date=August 6, 2003 |title=Smith, H.J. "Juggler Numbers" }}
*{{cite web|last1=Wright|first1=Colin|title=Juggling by Numbers|url=https://www.youtube.com/watch?v=7dwgusHjA0Y&t=17s|website=YouTube|publisher=[[Brady Haran]]|accessdate=4 October 2017|format=video}}
{{Juggling}}
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