What Is The Mathematics Behind Juggling

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WHAT IS THE MATHEMATICS BEHIND JUGGLING
Introduction
Claude Shannon is a computer scientist who is known for a well developed reputation in information theory. Claude Shannon is also known for “A Mathematical Theory of Communication” whereby, the field of Information Theory was created. Claude Shannon was also known an avid juggler, unicyclist and a tinkerer. Claude Shannon but a robotic juggling machine out of the parts of an Erector set and programmed out into a juggle three metal balls by making the bounce against a drum. In 1980 Claude Shannon published the first formal mathematical theorem of juggling by correlating the length of time balls in the air and how long each ball stays in the hand of the juggler. In this theorem, the importance of the speed of the hand to a successful juggling is also demonstrated. Juggling is a very interesting phenomenon that is practiced by people on their day to day life. Often time these individuals lack the knowledge to understand that there is mathematics behind the juggling of balls. Therefore mathematics has been fascinated by juggling since the development of this theory and this is why main aim of wanting to discover thus topics in mathematics. The following essay elaborates the mathematics behind the ball juggling.
Juggling comes down to a simple projectile motion whereby each ball follows a neat parabolic arc when tossed. However, multiple balls fallow interweaving paths in a periodically repetitive pattern. For a single juggler, there are three basic patterns that are used. These patterns include the cascade the fountain and the shower. A cascade is a pattern whereby odd number of balls is tossed from one hand to the other in two separate columns. A fountain is where even numbers are juggled in two separate columns also. Lastly a shower is a type of pattern in which all balls are juggled that is the even and odd number of balls. It is also important to note that an experience juggler might throw more than one object from a single hand at the same time. This kind of practice is known as multiplexing.
There are different combinations of throw, so the manner in which the juggler decides to produce variety of patterns depends on the site swaps. This means the mathematical notation of system referred to as the siteswaps assists a juggle to link each ball to how long a particular ball stays in the air. It is described in terms of beats. The availability of mathematical tools has assisted jugglers to see a particular pattern before attempting the same trick in the physical world. Juggling holds an aesthetic and an Intellectual appeal for the mathematicians.
A connection that exists between two attempts of a juggler to develop a kind of nation of their trick is called Site swap. This notation can be used to describe the pattern of juggling and it also turns out that standard mathematical tricks can also be used to develop new patterns that people are not aware of. Mathematician and juggler Colin Wright truest to explain the mathematics behind ball juggling using the following theorem. In this theory the basic unit of site swap is the throw. Every time a ball is usually thrown is represented using site swap as a number. For example, one of the most basic juggling patterns used is 3- ball cascade and each throw is usually represented as the number 2.
An example of an Illustration
A person can mix and match numbers to create a pattern. For example four ball patterns can be mixed up by swapping the places of the balls. If a person wants to exchange maybe the first ball with the second ball, the first ball can be thrown a little higher and the second ball a little lower so that they will be able to land where the other usually does. In a site swap notation, it means that throwing a 5 and a3 creates a new pattern known as 4-ball half shower.
One of the easiest ways to tell the pattern of a specific c site swap is valid is that if all the number of throws have to average out to the total number of the ball in that particular pattern. For example for a four-ball juggling, the average number of throw should be four. Fir a 4-ball fountain it is obvious because every throw is supposed to be 4. For half showers that number of throws is 5 and 3 which averages to 4. Additionally a person can go further instead of using 5 and 3, 6 and 2 which is pretty boring or 7 and 1 can also be used. Therefore a person can throw two throws as a 5, which means that the third throw should be a 2. Definitely this mathematics works out and it is a real pattern.
There are mathematical tricks that can be to discover new patterns. Wright used these tricks to invent 5551, which is a four ball pattern, and 441, which is a three ball pattern. No one had ever seen these tricks before, yet the math behind siteswap predicted them. Siteswap can get lots more complicated than this, and the math behind it gets more complicated too.
The Theory of Juggling
Shannon was also an accomplished juggler. He came up with the following elegant theorem, known as Shannon’s Juggling Theorem.
Shannon juggling equation is represented by
(F+D)H = (V+D) N
Breaking Down Shannon’s Equation (F+D) H = (V+D) NF is the time a ball spends in the air (Flight)
D is the time a ball spends in a hand (Dwell), or equivalently, the time a hand spends with a ball in it
V is the time a hand spends empty (Vacant)
N is the number of balls
H is the number of hands
The theorem can be derived by looking at a complete juggling cycle first from the perspective of the ball, then from the perspective of the hand, then equating the two times. This is an application of one of the most useful general tricks in combinatorics: double counting. A person can count and measure something in two different ways and use the fact that the two results have to be equal.
The equation of Shannon is used to illustrate the significance of the speed of hand of the Juggler. In this theory N increases when the juggler starts to lose the ability to vary the speed in the pattern while trying keep it stable. In this case H remains constant unless if another juggler comes into the pattern or the juggler sprouts more hands. Additionally, Shannon also developed the juggling robot from the Erector set. The juggling robot had two arms that could juggle up to three small metal balls against a drum. Over the past few years other engineers have also built robots that have the ability to bat objects upwards using the complex algorithms to make corrections. Moreover the websites of SARCOS also features a video of a humanoid robot that is juggling a three ball cascade pattern. This robot has cups of hands and it makes each toss and a catch in a smooth and fluid motion.
However, Kalvan also proposes an equation that can be used to describe the spatially optimal patter. This means that the equation allows the same amount of error of all points along the same flight path where the collisions are likely to occur. However, Kalvan’s equation is complicated and it concentrates on variables like the arc of each throw, finding the best distance between the arcs made from each hand and the ration that exists between when the hands hold the ball or when the hand remains empty.
An example of a juggling sequence 5,0,1