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. 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 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. 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 diagrams 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). See also Jugglers (category) Flair bartending References Further reading Dancey, Charlie 1995 Compendium of Club Juggling Butterfingers, Bath . Dancey, Charlie 2001 Encyclopedia of Ball Juggling, Butterfingers, Devon . Finnigan, Dave 1987 The Complete Juggler, Vintage Books, New York . Summers, Kit 1987 Juggling with Finesse, Finesse Press, San Diego . Ziethen, Karl-Heinz & Serena, Alessandro 2003 Virtuosos of Juggling, Renegade Juggling, Santa