Editing Juggling (section)

==Mathematics==
<!--[[Mathematics of juggling]] redirects directly here.--><!--[[Juggling theorem]] and [[Shannon's juggling theorem]] redirect directly here.-->
[[File:Juggling Shannon's theorem Cascade.png|thumb|An illustration of Shannon's juggling theorem for the cascade juggling pattern, note that the hand making the toss reverses each time through the pattern (1st time: RLR, 2nd time: LRL), meaning the tosses alternate between hands]]

Mathematics has been used to understand juggling as juggling has been used to test mathematics. The number of possible patterns ''n'' digits long using ''b'' or fewer balls is ''b''<sup>''n''</sup> and the average of the numbers in a siteswap pattern equal the number of balls required for the pattern.<ref name="Science"/> For example, the number of three digit three ball patterns is 3<sup>3</sup> = 27, and the box, (4,2x)(2x,4), requires (4+2+4+2)/4 = 3 balls.

"The time that a ball spends in flight is proportional to the square root of the height of the throw," meaning that the number of balls used greatly increases the amount of speed or height required, which increases the need for accuracy between the direction and synchronization of throws.<ref name = "Science"/>

[[Coupled oscillation]] and [[synchronization]] ("the tendency of two limbs to move at the same [[frequency]]"<ref name="Science"/>) appear to be easier in all patterns and also required by certain patterns. For example, "the fountain pattern...can be stably performed in two ways...one can perform the fountain with different frequencies for the two hands, but that coordination is difficult because of the tendency of the limbs to synchronize," while "in the cascade...the crossing of the balls between the hands demands that one hand catches at the same rate that the other hand throws."<ref name="Science"/>

{{multiple image
 | align = right
 | direction = vertical
 | width = 
 | header = Juggling ladder diagrams
 | image1 = Juggling - 3-ball cascade (3) ladder diagram.svg
 | caption1 = Cascade ladder diagram minimum required by siteswap (siteswap: 3)
 | image2 = Juggling - 3-ball cascade (3) ladder diagram Shannon.svg
 | caption2 = Cascade ladder suggested by Shannon's formula (siteswap: 3)
}}

[[Claude Shannon]], builder of the first [[juggling robot]], developed a '''juggling theorem''', relating the time balls spend in the air and in the hands: ''(F+D)H=(V+D)N'', where ''F'' = time a ball spends in the air, ''D'' = time a ball spends in a hand/time a hand is full, ''V'' = time a hand is vacant, ''N'' = number of balls, and ''H'' = number of hands.<ref name = "Science"/> For example, a hand's and a ball's perspectives in the two-hand (''H'') three-ball (''N'') cascade pattern:
 toss:    1st 2nd 3rd
 hand: D--VD—VD—V
 ball: D--F--D--F--
          R   L   R
          L   R   L
:(F+D)H=(V+D)N
:(3+3)2=(1+3)3
:6×2=4×3
:12=12

===Juggling notation===
[[File:Juggle.gif|thumb|right|With a few more balls: 10 siteswap]]
{{main|Juggling notation}}
Juggling tricks and patterns can become very complex, and hence can be difficult to communicate to others. Therefore, notation systems have been developed for specifying patterns, as well as for discovering new patterns.<ref>{{Cite news|url=http://thomwall.com/siteswap-fundamentals|title=Siteswap Fundamentals ⋆ Thom Wall|date=2017-09-05|work=Thom Wall|access-date=2017-11-21|language=en-US}}</ref>

Diagram-based notations are the clearest way to show juggling patterns on paper, but as they are based on images, their use is limited in text-based communication. Ladder diagrams track the path of all the props through time, where the less complicated [[causal diagram]]s only track the props that are in the air, and assumes that a juggler has a prop in each hand. Numeric notation systems are more popular and standardized than diagram-based notations. They are used extensively in both a written form and in normal conversations among jugglers.

[[Siteswap]] is by far the most common juggling notation. Various heights of throw, considered to take specific "beats" of time to complete, are assigned a relative number. From those, a pattern is conveyed as a sequence of numbers, such as "3", "744", or "97531". Those examples are for two hands making alternating or "asynchronous" throws, and often called ''vanilla siteswap''. For showing patterns in which both hands throw at the same time, there are other notating conventions for synchronous siteswap. There is also ''multiplex siteswap'' for patterns where one hand holds or throws two or more balls on the same beat. Other extensions to siteswap have been developed, including passing siteswap, Multi-Hand Notation (MHN), and General Siteswap (GS).