Mean ball height
Due to the potential demands of this first section, this section will be done as a class led by the instructor.
Juggling is combination of an athletic art and science. Underneath all juggling patterns are mathematical models that determine whether or not a particular pattern can or cannot be juggled.
The instructor will demonstrate a left-right inner throw cascade.
The balls always follow a parabolic path in the air. The flight time for a ball is equal to twice the time to fall from the top of the ball's parabolic arc. The time to fall can be calculated from the general formula:
Mean ball height
Where h is the mean height, a is the acceleration, and t is the time. Solving for t and substituting the acceleration of gravity g for a yields:
The total time in the air for the ball - the time to rise to the top of the arc and fall back into the hand is twice the fall time:
The time that a ball rests in the hand between a catch and a throw is the dwell time d. There is also a brief period of time when the hand is empty called the empty time e. The hand period is the dwell time plus the empty time and is usually denoted the tau time τ.
Claude Shannon's juggling theorem states that where b is the number of balls, and h is the number of hands, and f is the total flight time 2t, then:
In our lab today balls = 3 and hands =2. To determing the flight time we will time 12 throws counting throw from both hands. During the twelve throws as counted from both hands, one ball shuttles four times back and forth in front of the juggler. This time includes flight, dwell, and empty times. Divide the time for 12 throws from both hands by 4 to calculate the period p. We can calculate the flight time from the mean height using the formula above. The period p minus the flight time is equal to the dwell time + the empty time. That is p − f = d + e
d + e is an important measure of an efficient juggler. d + e is also called the hand period tau τ. τ is a known value for some professional jugglers.
| # balls | preferred τ |
|---|---|
| 3 | 0.47- R. Rubenstein (1994) 0.54 - J. Kalvan (1994) 0.62 - B. Olson (1994) |
Use the following table to stay organized in calculating your own τ value:
[This table is known to be in error as of December 2007. The 12p period was counts of both hands, twelve left and right throws. Six right hand throws. Using time-space diagrams this means 6 × (d+e) = the "12p" time. The table co-mingles the hand and ball perspective. If this lab is done spring 2008, this should be sorted out. While the Shannon's theory work was interesting, ultimately it was more confusing than fun. ]
| mean height μ | g | flight time f | time for twelve throws | 12/4 period p for one ball | τ = d + e (=p − f) |
|---|---|---|---|---|---|
| 9.79 |
Knowing that b/h is 1.5, attempt to solve for and calculate your e value using Shannon's theorem and the values in the table above.
There is a mathematical notation for juggling that describes the possible juggling patterns for a given number of balls. The notation is called site swap notation, alternatively written as a single word siteswap. The numbers are an abstract short hand. The numbers do not tell you how to juggle, the numbers do not tell you where to move your hands, nor whether to throw balls over or under other balls.
A simple three ball cascade juggle has each of the three balls passing through each of the hands in alternation. Each hand must touch each ball in succession, with the balls being passed from the right to the left and back to the right. The first ball thrown must "fill" the right hand after the third ball has left the right hand.
In the above space-time diagram time is passing as one moves from left to right.
If we "interleave" the left hands inbetween the right hands thusly:
...we obtain the following diagram:
Studying the above diagram we can see that the ball moves three "hands" downrange before landing on the other hand (think of hands as landing sites, hence the use of the word site in the name of the notation). The first ball returns to the right hand after moving three more hands (or sites) rightward in the diagram.
There are three basic patterns in juggling.
In the cascade each ball follows this exact same symmetrical pattern. Mathematically this is recorded as 3 3 3 3 3 3 3 3 3... To save space, we toss the spaces and write 33333... For further simplicity we invoke a rule that number patterns repeat indefinitely, hence a single 3 can be used to denote this pattern.
Another pattern is the three ball shower, with the balls always being thrown from one hand and then caught and fed back to the first hand in a transfer. The right hand must "launch" the balls in sequence "downrange" to the opposite hand. The first opposite open "site" (hand) that permits a transfer (a throw of 1) to set up the relaunch of ball one is the fifth site.
Mapping the launches and landings of the other two balls confirms that the balls land "5 site" downrange. This is refered to as a 5 1, or 51 with the spaces removed, with each ball executing the same 51 pattern over and over: 515151...
Other notes:
The notation is called site swap in part because one can "swap" or "switch around" the "sites" (hands!) on which balls land. That it, from a three ball cascade one can toss a ball from the right hand straight up, catching the ball four sites (or four counts or four beats) later. This gives one time to catch and throw a ball from the left hand while the 4 count ball is airborne. While this is happening the left hand holds onto a ball. This is counted as a "2" in the world site swaps (even numbers return to the same hand, holding a ball for an extra beat results in calling this a count of 2).
The diagram depicts a start of a three ball cascade with a single column ball off the right hand followed by a return to a three ball cascade. The notation, unsimplified, would be 33423333...
To determine the number of balls calculate the average for each sequence. Add the numbers as single digits and then divide by the number of numbers. [Note to real site swappers: this is intended as math exercise and the author does not mean to imply that this condition is by any means sufficient. It is not sufficient. The class will not be checking for multiplex landings, transition issues, or excited states.]
Determine the number of balls in the following patterns. If the mean is fractional, then the pattern does not exist or is incomplete.
Running and juggling is difficult with anything other than patterns that synchronize with the cadence of the runner. Assymetrical patterns such as 51 are very difficult to maintain while running. Since cadence is essentially a two beat, left right proposition, this author suspects that only patterns of single digits N M where N=M are really functional. This is potentially a massive limitation leaving only simple cascades as a functional possibility.
For more information and further reading see the following pages:
Siteswap FAQ by Allen Knutson
Siteswap from Wikipedia
The Invention of Juggling Notations
The science of juggling
| Sect | mean height h/cm | g | flight time f | 12p | d+e+f | τ = d + e | d | e |
|---|---|---|---|---|---|---|---|---|
| Th08 | 65 | 979 | 0.7288 | 4.25 | 1.0625 | 0.03337 | ||
| Th11 | 60 | 979 | 0.69 | 4.75 | 1.1875 | 0.5 | 0.06 | 0.44 |
[The above table is thought to be in error, hence the negative empty time that the 8:00 lab obtained. The column d+e+f should be simply d+e, maybe.]