By Burkard Polster
ISBN-10: 0387227482
ISBN-13: 9780387227481
ISBN-10: 0387955135
ISBN-13: 9780387955131
Learn to juggle numbers! This ebook is the 1st complete account of the mathematical innovations and effects utilized in the modelling of juggling styles. This comprises all identified and lots of new effects approximately juggling sequences and matrices, the mathematical skeletons of juggling patterns.
Many precious and enjoyable tips and methods boost the mathematical menu awarded during this booklet. There are specific descriptions of jugglable and tasty juggling sequences, effortless zero-gravity juggling, robotic juggling, in addition to enjoyable juggling of phrases, anti-balls, and irrational numbers.
The e-book additionally comprises novel, or a minimum of no longer rather well identified connections with subject matters reminiscent of bell ringing, knot idea, and the various physique challenge. in truth, the bankruptcy on mathematical bell ringing has been improved into the main accomplished survey within the literature of the maths utilized by bell ringers.
Accessible in any respect degrees of mathematical sophistication, it is a booklet for mathematically stressed jugglers, mathematical bell ringers, combinatorists, arithmetic educators, and as regards to anyone attracted to attractive and weird functions of mathematics.
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Extra resources for The Mathematics of Juggling
Example text
Clearly, since φs is a permutation, so is φs . We summarize the results of this subsection as follows: 28 2. Simple Juggling The Inverse of a Juggling Sequence Let s = {ak }p−1 k=0 be a juggling sequence, and let φs be its associated permutation. Then, its inverse is the juggling sequence s = {aφ−1 }p−1 s (p−1−k) k=0 and φs (k) = p − 1 − φ−1 s (p − 1 − k) for k = 0, 1, . . , p − 1. 3 Pick a Pattern Procedure In [94], page 168, Martin Probert gives a recipe for constructing some of the b-ball juggling sequences of period p that boils down to the following.
In fact, it turns out that this group is the affine Weyl group A˜p−1 . Two juggling sequences of period p correspond to the same element of A˜p−1 if and only if one is the vertical shift of the other; see page 23 for a definition of the term vertical shift. 7 How Many Ways to Juggle? 43 Clearly, A˜0 is the one-element group. For p ≥ 2, it can be shown that A˜p−1 is generated by the simple reflections ⎧ ⎨ i + 1 for i mod p = k mod p, i − 1 for i mod p = (k + 1) mod p, tk : Z → Z : i → ⎩ i for i mod p = k, (k + 1) mod p, for k = 0, 1, .
If the number is a 0, choose a copy of card C0 and place it above the 0. Choose any of the numbers that are not 0, say the 1 (boxed in the diagram). 40 2. Simple Juggling From the circle that indicates the corresponding beat, trace backwards on the incoming arc and inspect the crossings with other arcs. When we cross another arc, we either cross from the inside to the outside or in the other direction. In the first case, we mark the crossing by a gray circle, otherwise by a gray square. Count the number c of gray circles.
The Mathematics of Juggling by Burkard Polster
by Paul
4.2