By Burkard Polster

Learn to juggle numbers! This ebook is the 1st finished account of the mathematical thoughts and effects utilized in the modelling of juggling styles. This comprises all recognized and lots of new effects approximately juggling sequences and matrices, the mathematical skeletons of juggling patterns.

Many valuable and interesting information and tips boost the mathematical menu provided during this booklet. There are exact descriptions of jugglable and engaging juggling sequences, effortless zero-gravity juggling, robotic juggling, in addition to enjoyable juggling of phrases, anti-balls, and irrational numbers.

The booklet additionally comprises novel, or a minimum of no longer rather well recognized connections with issues similar to bell ringing, knot concept, and the various physique challenge. in reality, the bankruptcy on mathematical bell ringing has been increased into the main entire survey within the literature of the math utilized by bell ringers.

Accessible in any respect degrees of mathematical sophistication, it is a ebook for mathematically stressed out jugglers, mathematical bell ringers, combinatorists, arithmetic educators, and near to anyone attracted to attractive and strange purposes of mathematics.

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**Extra resources for The Mathematics of Juggling**

**Example text**

20, this state is marked by a gray rectangular box. The shortest closed path in any state graph starts and ends at the ground state and consists of the only loop-edge in the graph. The corresponding juggling sequence is the basic one-element juggling sequence b. All states diﬀerent from the ground state are called excited states. 20: 3 42 342 531 5340 55500 Note that any two juggling sequences that originate at the same state can be concatenated to form another juggling sequence that originates at the same state.

Apart from some trivial exceptions, ﬁxing the number of balls and maximum throw height will give an inﬁnite class of juggling sequences. 8 that a lot of interesting results can be proved about the b-ball juggling sequences that consist of throws up to a certain ﬁxed maximum height. This includes, among other things, very eﬀective ways of listing all such sequences of a given period p by ﬁnding the number of loops in a certain graph. Of course, this is exactly what we are interested in when limiting all three parameters.

Hence, we are really dealing with a juggling sequence. Concatenating two b-ball juggling sequences of periods p1 and p2 constructed like this yields a b-ball juggling sequence of period p1 + p2 that also arises in this manner. 16 should be enough to tell you why this is so. 16. Joining two “picked” b-ball juggling sequences yields a new picked b-ball juggling sequence. In this example, the sequences 42 and 333 combine into the new sequence 42333. How many b-ball juggling sequences of period p can be produced with this method?