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 [[positivenon-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|lastlast1=Beek|firstfirst1=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>

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"/>

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>

This pattern can be describedescribed by stating how many throws later each ball is caught. For instance, on the first throw in the diagram, the purple ball is thrown in the air (up out of the screen, towards the bottom left) by the right hand, next the blue ball, the green ball, the green ball again, and the blue ball again and then finally the purple ball is caught and thrown by the left hand on the fifth throw, this gives the first throw a count of ''5''. This produces a [[sequence]] of numbers which denote the height of each throw to be made. Since hands alternate, [[parity (mathematics)|odd]]-numbered throws send the ball to the other hand, while even-numbered throws send the ball to the same hand. A ''3'' represents a throw to the opposite hand at the height of the basic three-[[cascade (juggling)|cascade]]; a ''4'' represents a throw to the same hand at the height of the four-[[fountain (juggling)|fountain]], and so on.

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 ( )|parentheses]] and separated by a comma. Since synchronous throws are only thrown on even beats, only even numbers are allowed.<ref name=":1">{{Siteswap Ben's|page=6}}</ref> Throws that move to the other hand are marked by an ''x'' following the number. Thus a synchronous three-prop [[shower (juggling)|shower]] is denoted ''(4x,2x)'', meaning one hand continually throws a low throw or 'zip' to the opposite hand, while the other continually makes a higher throw to the first. Sequences of bracketed pairs are written without delimiting markers. Patterns that repeat in mirror image on the opposite side can be abbreviated with a *. For example, Instead of ''(4,2x)(2x,4)'' (3-ball [[Box (juggling)|box]] pattern), can be abbreviated to ''(4,2x)*''.

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.)

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, 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.

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|url-status=live|archive-url=|archive-date=|access-date=April 22, 2020}}</ref>

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, Thethe results from the previous example would produce <math>5\bmod 3,4\bmod 3,3\bmod 3</math>or <math>2,1,0</math>. In this case, 531 is valid since the numbers <math>2,1,0</math> are all unique. Another example, 513 is an invalid pattern because the first step produces <math>5+0,1+1,3+2 </math> or <math>5,2,5</math>, the second step produces <math>5\bmod 3,2\bmod 3,5\bmod 3</math> or <math>2,2,2</math>, and the final sequence contains at least a duplicate of one number, in this case a 2.

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.

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 1/3≠31≠3, 4/3≠31≠3, (4+1)/3≠32≠3, and (4+4)/3≠32≠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|url-status=live|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].

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|lastlast1=Ehrenborg|firstfirst1=Richard|last2=Readdy|first2=Margaret|date=1996-10-01|title=Juggling and applications to q-analogues|url=http://www.sciencedirect.com/science/article/pii/S0012365X9683010X|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'') &minus; ''i'': that is, the ball thrown at time ''i'' will land at time ''f''(''i'').

*[httphttps://wwwydgunz.gunswapgithub.coio/gunswap/ Gunswap] - A web based, open source, 3d juggling animator and pattern library.

*[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)