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- The Mathematics of Juggling.
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As a result of this research, many new possibilities have been discovered for jugglers to attempt. In addition, the connections between juggling and the algebra of braids provide another way to analyze juggling. More videos from this series. The late computer scientist Claude Shannon has a well-deserved reputation as the father of information theory, but he was also an avid unicyclist, juggler and tinkerer. He even built his own robotic juggling machine out of parts from an Erector set, programming it to juggle three metal balls by bouncing them against a drum.

His theorem demonstrated the importance of hand speed to successful juggling. Mathematicians have been fascinated by juggling ever since. In essence, juggling comes down to simple projectile motion, with each ball following a neat parabolic arc as it is tossed — except that there are multiple balls following interweaving paths in periodically repeating patterns.

For a single juggler, there are three basic patterns: A more experienced juggler might throw more than one object from a single hand at the same time, a practice known as multiplexing. There are many possible combinations of throws, so how do jugglers decide which ones will produce a valid pattern? For example, a one-beat throw means the juggler simply passes the ball from one hand to the other.

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Choose a sequence from the list on the left, and then watch a virtual juggler perform it for you. Click on the juggler to stop and start the animation. Moving the mouse around while clicking on the juggler will rotate the view of the juggler. All the juggling sequences mentioned in this article, and many more, are listed.

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I recommend playing with this simulator for a while before reading on. A standalone program, with many more features, can be downloaded here.

## The Mathematics of Juggling

To discover what is possible in terms of simulating multiple jugglers manipulating chainsaws while riding unicycles in 3D, have a play with JoePass! Back to John, who's trying to figure out whether his favourite sequence 5,0,1 is a juggling sequence "favourite" because John is also a " darts" fan. To do this he draws a diagram — on a library window, of course. This diagram consists of a row of equally spaced dots, representing the beats on a timeline, and labelled with the corresponding throw durations. A dot labelled with 0 means no ball is thrown on that beat. From each dot with a positive number he draws an arc, extending that number of dots to the right.

There is now a simple test of whether the diagrammed sequence is a juggling sequence: Also, for a juggling sequence, the arcs join to form exactly as many curves as there are balls being juggled. This all tells us, for example, that 5,0,1 is jugglable and that we need two balls to juggle it. In order to juggle 1,2, John would have to catch two incoming balls in one hand on every second beat. Moreover, John would need to have balls mysteriously materialise and vanish. None of this is possible within our simple juggling formulation, and therefore 1,2 is not a juggling sequence.

Suppose we want to determine whether 4,4,1,3 is jugglable. We create a second sequence, by adding 0,1,2,3 to the elements of the first sequence. This gives 4, 5, 3, 6. Then buckle your seat belts , we form a third sequence whose elements are the remainders of the elements of the second sequence when divided by the length of the original sequence.

Our sequence has length 4, and so the new sequence is 0,1,3,2. It turns out that the original sequence is a juggling sequence exactly if this final sequence consists of distinct numbers. That is true for the sequence 0,1,3,2, and so 4,4,1,3 really is a juggling sequence.

## Juggling, maths and a beautiful mind

The monster ,1,1 is as well, as you can check. You can find out why this works in the articles and books listed below. Also, it turns out that the number of balls necessary for juggling a sequence is simply its average. Usually jugglers write instead of 4,4,1,3. This makes sense, as long as it is understood that all the digits stand for different throws. To be able to record throws of greater than nine beats it is common to let A stands for 10, B for 11, and so on.

Of course, to a playful mathematical mind this immediately invites one to check whether their name is jugglable. In fact, people have checked all the words in the dictionary for jugglability. It turns out that very few real words are jugglable, though curiously both "theorem" and "proof" are. Which do you think is harder, to come up with a conjecture for a theorem or to prove it?

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Well, our virtual juggler thinks that "proof" is harder than "theorem", because it takes 23 balls to juggle the first word, whereas "only" 21 are required to juggle the second. The diagram above contains all three ball juggling sequences with throws that last at most 5 beats. Here is how we extract them. Starting at any of the rectangles, we follow any path of arrows, recording the numbers in the orange circles along the way.

Once we return to the beginning rectangle, the string of recorded numbers is a juggling sequence. For example, beginning at the brown rectangle and walking along the red, green, and purple loops produces the juggling sequences 3 and 4,4,1 and 5,5,1,5,0,5,3,0. We can also combine the red and green loops into a longer loop that visits the brown rectangle twice, giving the juggling sequences 4,4,1,3.

## Juggling, maths and a beautiful mind | ibenimoq.tk

Importantly, even if you are familiar with the two patterns 4,4,1 and 3, the combined pattern 4,4,1,3 will look and feel completely different. This means that on the next five beats there are scheduled to be 1, 0, 1, 1, 0 balls landing. So, I am currently in the juggling state This means that on the following five beats there are 0, 1, 1, 0, 0 balls scheduled to land. Consequently, if I want to avoid collisions, I can only choose between throws of 1, 4, and 5 beats duration, corresponding to the positions of the 0s in this last sequence. If I decide on a 4 beat throw, then I have to replace the 0 in the fourth position by a 1 and my new juggling state becomes If I go for a 1 beat throw, my juggling state changes to , and if I perform a 5 beat throw my state becomes This calculation demonstrates the connection between the original state and the three arrows leading from it to the states , and Repeating the same for all possible states gives the complete juggling state graph.