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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 [[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 journal|last1=Beek|first1=Peter J.|last2=Lewbel|first2=Arthur|title=The Science of Juggling|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}}</ref>
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".
==Vanilla==
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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
!Throw Name
!Beats object is in Air
!Switches hands
!Description
|-
|0
| -
| -
|Empty hand
|-
|1
|1
|Yes
|Throw from one hand to the other
|-
|2
|0
|No
|Momentary hold
|-
|3
|3
|Yes
|Throw from a 3 ball [[Cascade (juggling)|cascade]]
|-
|4
|4
|No
|Throw from a 4 ball [[Fountain (juggling)|fountain]]
|-
|5
|5
|Yes
|Throw from a 5 ball [[Cascade (juggling)|cascade]]
|-
|6
|6
|No
|Throw from a 6 ball [[Fountain (juggling)|fountain]]
|-
|7
|7
|Yes
|Throw from a 7 ball [[Cascade (juggling)|cascade]]
|-
|8
|8
|No
|Throw from a 8 ball [[Fountain (juggling)|fountain]]
|-
|9
|9
|Yes
|Throw from a 9 ball [[Cascade (juggling)|cascade]]
|-
|a
|10
|No
|Throw from a 10 ball [[Fountain (juggling)|fountain]]
|-
|b
|11
|Yes
|Throw from a 11 ball [[Cascade (juggling)|cascade]]
|-
|...
|...
|...
|...
|}
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.
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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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Simultaneous juggling: ''<xxx|yyy>'' notation means one juggler does 'xxx' while another does 'yyy'. 'p' is used to represent a passing throw. For example, ''<3p 3|3p 3>'' is a 6 prop '2 count' passing pattern, where all left hand throws are passes and right hand throws are selves. This can also be used with synchronous patterns; a two-person 'shower' is then ''<(4xp,2x)|(4xp,2x)>''
===Fractional notation===
If the pattern contains fractions, e.g. ''<4.5 3 3 | 3 4 3.5>'' the juggler after the bar is supposed to be half a count later, and all fractions are passes.
'''social siteswap'''
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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In practice, this is most easily obtained if the jugglers throw by turns, one sequence being (Right hand of juggler A, right hand of juggler B, left hand of A, left hand of B).
'''mixed-up notation'''
Some jugglers, when noting 4-handed siteswap, divide the siteswap values by the number of jugglers. This leads to a fractional notation similar to the notation for social siteswaps, but the order of the notation can be different.
==State diagrams==
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A siteswap state diagram can also be represented as a [[state-transition table]], as shown on the right. To generate a siteswap, pick a starting state row. Index into the row via the corresponding throw column. The state entry at the intersection is the transitioned to state when that throw is made. From the new state, one can index into the table again. This process can be repeated so that when the original state is reached, a valid siteswap will be created.
==Mathematical
===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
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
# it only contains even numbers and
# it can be converted into a valid vanilla siteswap using the {{Pslink|
otherwise it is invalid{{Citation needed|date=April 2020}}.
=== Swap
New valid
For example, the swap property performed on the inner two throws of the sequence 4413 would move the 4 to the right subtracting 1 from it to become 3 and move the 1 to the left adding 1 to it to become 2. This produces the new valid siteswap pattern 4233.
=== Slide
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 1/1≠3, 4/1≠3, (4+1)/2≠3, and (4+4)/2≠3). 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 <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|archive-url=|archive-date=|access-date=April 27, 2020}}</ref> The longest prime patterns with height 22 (with 3 ball maximum), for 9 balls (with 13 maximum height), and for heights and ball counts in between, were enumerated by Jack Boyce in February 1999 using a program called jdeep.<ref>{{Cite web|url=http://www.sonic.net/~boyce/jdeep5.c|title=jdeep.c|last=Boyce|first=Jack|date=February 17, 1999|website=sonic.net|url-status=dead|archive-url=https://web.archive.org/web/20041207120815/http://www.sonic.net/~boyce/jdeep5.c|archive-date=December 7, 2004|access-date=April 27, 2020}}</ref> The full list of longest prime siteswaps generated by jdeep (with '0' throws represented by a '-' and maximum height throws represented by a '+') can be found [https://web.archive.org/web/20110317025440/http://www.jugglingdb.com:80/compendium/boyce/prime_list.html here].
== Mathematical connections ==
==
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|last1=Ehrenborg|first1=Richard|last2=Readdy|first2=Margaret|date=1996-10-01|title=Juggling and applications to q-analogues|journal=Discrete Mathematics|language=en|volume=157|issue=1|pages=107–125|doi=10.1016/S0012-365X(96)83010-X|issn=0012-365X|doi-access=free}}</ref> One presentation of this group is as the set of [[bijective]] functions ''f'' on the integers such that, for a fixed ''n'': ''f''(''i'' + ''n'') = ''f''(''i'') + ''n'' for all integers ''i''. 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 ''i''th number is ''f''(''i'') − ''i'': that is, the ball thrown at time ''i'' will land at time ''f''(''i'').
=== Connections to
A subset of these siteswap patterns naturally label strata in the positroid stratification of the [[Grassmannian]].<ref>{{Cite arXiv|last1=Knutson|last2=Lam|last3=Speyer|first1=Allen|first2=Thomas|first3=David|eprint=1111.3660|title=Positroid Varieties: Juggling and Geometry|date=November 15, 2011|class=math.AG}}</ref>
==List of symbols==
{{Div col}}
*Number: Relative duration (height) of a toss. 1, 2, 3...
*Brackets []: Multiplex. [333]33.
*Chevrons and vertical bar <|>: Simultaneous and passing patterns.
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There are many free [[computer program]]s available which [[simulation|simulate]] juggling patterns.
*[https://jugglinglab.org/ Juggling Lab animator] - An [[open source]] animator which was written in [[Java (programming language)|Java]] and interprets nearly all siteswap syntax.
*[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)
*[https://ydgunz.github.io/gunswap/ Gunswap] - A web based, open source, 3d juggling animator and pattern library.
There are also some games to play with siteswap:
*[https://www.dropbox.com/s/y2jse1mdzqatcrz/SiteswapGame.rar?dl=0 Siteswap Game] developed by Sebi Haushofer (for Java)
==See also==
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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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