How to juggle the proof for every theorem An exploration of the mathematics behind juggling Mees Jager 5965802 (0101) 3 0 4 1 2 (1100) (1010) 1 4 4 3 (1001) (0011) 2 0 0 (0110) Supervised by Gil Cavalcanti Contents 1 Abstract 2 2 Preface 4 3 Preliminaries 5 3.1 Conventions and notation . .5 3.2 A mathematical description of juggling . .5 4 Practical problems with mathematical answers 9 4.1 When is a sequence jugglable? . .9 4.2 How many balls? . 13 5 Answers only generate more questions 21 5.1 Changing juggling sequences . 21 5.2 Constructing all sequences with the Permutation Test . 23 5.3 The converse to the average theorem . 25 6 Mathematical problems with mathematical answers 35 6.1 Scramblable and magic sequences . 35 6.2 Orbits . 39 6.3 How many patterns? . 43 6.3.1 Preliminaries and a strategy . 43 6.3.2 Computing N(b; p).................... 47 6.3.3 Filtering out redundancies . 52 7 State diagrams 54 7.1 What are they? . 54 7.2 Grounded or Excited? . 58 7.3 Transitions . 59 7.3.1 The superior approach . 59 7.3.2 There is a preference . 62 7.3.3 Finding transitions using the flattening algorithm . 64 7.3.4 Transitions of minimal length . 69 7.4 Counting states, arrows and patterns . 75 7.5 Prime patterns . 81 1 8 Sometimes we do not find the answers 86 8.1 The converse average theorem . 86 8.2 Magic sequence construction . 87 8.3 finding transitions with flattening algorithm . 88 8.4 A lot more to be said about prime patterns . 88 9 Of interest to the juggler 90 9.1 The basics . 90 9.2 Changing, constructing and decomposing sequences . 91 9.3 Transitions . 92 10 Some final remarks 96 10.1 The impact on juggling in general and myself . 96 10.2 About the title . 97 11 Acknowledgements 98 12 Appendix 99 1 Abstract As the subtitle clarifies this thesis is on the mathematics behind juggling. Whenever we want to learn, talk about or describe something, we need a lan- guage which fits the subject well. Sometimes the language we speak suffices and all we need is to find some common ground by developing terminology. However, sometimes this does not suffice and we need to develop a language to fit the subject, taking music as an example. Imagine having to commu- nicate a piece of music without having a dedicated language and notation, it's very hard! A similar thing is to say for communicating juggling so a language was developed. Given the precise nature of juggling, it is not much of a surprise that this became a mathematical language. Having a mathematical language means having definitions and we need to check that these capture our phenomena we try to describe. We do so by making sure that, given our abstract definition of a juggling pattern, we can retrieve all the information necessary to actually juggle the pattern in the detail we are aiming to describe it. We answer practical questions like how many balls a pattern requires and where we need to start in the pattern. 2 Having a mathematical language also allows very well for seemingly non- sensical questions to be asked. Since we can be sure that we translated juggling into a piece of mathematics which makes sense for our topic, we can forget about the juggling for a moment and play around with just the mathematics. The theorems we then prove might not have any interesting physical interpretation at all, but sometimes we can relate these back to very nice properties of the juggling sequences. This thesis tries to explore all of these facets. It should be viewed as a collection of theorems and proofs, which start rudimentary and build up to more technical results, which are hopefully interesting and useful for math- ematicians and jugglers alike. 3 2 Preface "Jugglers and mathematicians share the joy of discovering new patterns" - Anneroos Everts A few words on why I would write a mathematics thesis on juggling. The quote above captures the connection between the two fields very well. I am a juggler myself and already knew a fair bit on the notation and theory, but had a lot of gaps in this knowledge. It was a lot of fun to fill those in and get a better understanding and new found appreciation of my hobby. Personally, I learned a lot and I especially enjoyed writing on and learning about state diagrams, because it allowed me to think about a question I had in the back of my mind for a long time already, but never urged myself to think about. This would be "how can we find transitions between patterns in a more prac- tical sense?" and I believe that I gained a lot of insight in this, which I hope you find as entertaining to read about, as I find it was to think about. The way in which I wrote the proofs and structured the whole thesis is with an abundance of detail. I find myself often annoyed by how little de- tail is given in textbooks and how easily the words "clearly", "obviously", "trivially" etc. are used. This is personal of course, but as a thesis is quite personal as well, I took this opportunity to steer away from this and write in the way I would like to read it, which is with a lot of detail. I should admit that I also just struggle with being concise. The danger of this is writing too much detail, which I am sure many people will feel I have done if they would read it, and I can even imagine myself looking back on this in some years, being annoyed with how much detail I used. However, for now it feels right, and I do hope whoever might read this, finds it enlightening and the details helpful. 4 3 Preliminaries 3.1 Conventions and notation We will use modular arithmetic a lot throughout this thesis and we would like to distinguish between evaluating some integer modulo p and saying that two integers are equivalent modulo p. Definition 3.1. We define rp : Z ! Z=pZ to be the remainder function, which means that rp(a) is the remainder of a after division by p. So a = rp(a) + kp for some k 2 Z and 0 ≤ rp(a) < p. Notation. When we write a ≡ b (mod p) we mean that a = b + kp for some k 2 Z. Here we do really mean that a and b belong to the same class (mod p). We will be working with the set f0; 1; ::; p − 1g a lot so we want to have some notation for this. Notation. Let Z=pZ denote the set f0; 1; ::; p − 1g. Even though Z=pZ in- herits a ring structure as an abstract quotient, and we will call on this from time to time by adding elements modulo p, we really want to see Z=pZ as a set of these integers and not see the elements as representatives of their class. p−1 p−1 Notation. Let (ai)i=0 and (bi)i=0 be sequences of integers. We will write p−1 p−1 rp((ai)i=0 ) to denote the sequence (rp(ai))i=0 where we evaluate every element p−1 p−1 modulo p. In this manner we also write (bi)i=0 ≡ (ai)i=0 (mod p) if bi ≡ ai (mod p) for all i 2 Z=pZ. 3.2 A mathematical description of juggling We want to study juggling patterns and introduce mathematics into them and in order to do so we need to disregard some irrelevant aspects of a pat- tern and also make some assumptions. We will use this section to make clear what these assumptions are, give ourselves a feel for what we mean with juggling patterns and then abstractly define what a juggling pattern is. First of all we ignore the objects a juggler uses and call them balls from now on. Furthermore we can safely assume that the number of balls stays 5 constant during a juggling pattern and we disregard the path that a ball takes in the air, i.e. whether it goes in front of or behind the body or whether it goes underneath a leg or not etc. We can go on to disregard the number of jugglers involved in a pattern and even the number of hands involved. However, we do assume that, whatever the number of hands involved is, we decide on an order in which they throw and stick to this throwing order dur- ing the pattern. It will be convenient to say that the pattern has been going on forever and will continue to do so. We assume the throws in a pattern to happen at a constant rhythm and that a pattern as a whole is periodic. This constant rhythm means that throws happen at discrete equally spaced moments in time and we can use this to interpret a throwing moment t, from now on called a beat, as an integer t 2 Z. We assume that, at every beat, either one or no balls land and once a ball is caught, it is thrown again imme- diately and if no balls land no throw is made at that beat. This last remark could also be seen as the requirement that no two or more balls thrown or caught at the same time, which, among jugglers, would be referred to as a "squeeze catch" or "multiplex throw" respectively.