Published September 2025 as part of the proceedings of the first Alpaca conference on Algorithmic Patterns in the Creative Arts, according to the Creative Commons Attribution license. Copyright remains with the authors.
doi:10.5281/zenodo.17084412
Jugglers and mathematicians share the joy of discovering new patterns and it is no surprise that there is a deep connection between the two. The patterns jugglers make can be described with a mathematical system commonly known as siteswap. It offers jugglers a way to communicate about the patterns they create. Furthermore, this theoretical description has resulted in the findings of new juggling patterns that would not have been so easy to cook up by simply trying to juggle something new. In that sense, it is not only a jugglers language, but also a jugglers toolbox.
Siteswap describes a juggling pattern by a finite sequence of numbers. It is common to also call such a sequence of numbers, and therefore a juggling pattern itself, a siteswap. To a juggler, a siteswap is the equivalent of sheet music to a musician. It is their text so to speak. Every number represents a throw and has meaning to the juggler in the sense that it tells them roughly how high and in what direction to make the corresponding throw. As the juggler reads the numbers they make the throws in the order they read them. Once you get to the end of the sequence you start from the beginning again.
Not every sequence of numbers is a juggling pattern though, as it needs to obey the rules of siteswap. These rules, or assumptions if you will, are necessary to make the theory work.
This talk aims to convey an overview of siteswap, starting at the basic definition and going as far as being able to work out how to juggle such a string of numbers. Given enough practice of course. In a combined workshop we aim to get our hands dirty and use this freshly learned knowledge to actually learn how to juggle ourselves in a much more sophisticated way than by simply brute forcing some throws. Indeed, understanding the theory behind it will help you to learn juggling a lot faster and a lot better, or so I claim. It is important to note that you do not need to be a juggler or a mathematician in order to attend and hopefully enjoy this talk. All you need is a fascination for patterns, which, if I understand the conference correctly, is just about everyone attending the festival.
In order for this system to work we need to make several assumptions about the juggling patterns we aim to describe. We define the moments at which a juggler makes a throw as a beat.
Let’s give some context to these assumptions. The first can be thought of as juggling on a metronome, only making throws at the clicks. The second one simply states that once the right hand made a throw, the left hand must make a throw, then the right again etc. Assumption 3, 4 and 5 sound very obvious but we do need them to make sense of all this. No balls can land at the same time, once a ball lands we assume it is thrown that very same beat and if no ball lands at a beat then no throw is made. Assumption 6 tells us that juggling must be periodic.
Ideally this talk covers all of the below topics, but due to time restrictions this is not realistic. The main focus of this talk will be to give the listener a rough understanding of what the theory entails, what topics there are to further dive into and to give a small preparation for those interested in the corresponding workshop. All of this with plenty of real life juggling examples to go with the theory.
We will first focus on explaining the system through examples. The goal here is to understand what a number means in terms of juggling and how only a string of numbers can tell a juggler how many balls they need and where and how high to throw the balls
We move on to describing how to construct new patterns our of existing ones in various ways. We also take a different viewpoint; if siteswap tells us what the juggler does at every beat, what happens if we focus on a single ball and follow its path in the pattern? This is where orbits start to play a role. An orbit is dependent on a ball and tells us what each ball does at every beat. Multiple balls can share the same orbit and knowing all the different orbits there are in a pattern is enough to reconstruct the pattern as a whole.
Finally, we hope to spend some time on discussing state diagrams. To understand what a state is, imagine the following; say we are juggling some pattern and stop making throws at some point. Then no balls are being launched anymore but they are still coming down. Due to the rules of siteswap we know that at every beat either 1 ball or 0 balls are landing. We record this for every beat after we stopped juggling with a respective 1 or 0. We now have a sequence of 1’s and 0’s telling us in what order balls are coming down after we stop juggling the pattern at a specific point. This sequence is what we call the state of the pattern at the throw where we decided to stop juggling. If we were to stop juggling one throw later the state would be different from the one we just obtained.
Having information about the different states your pattern has, is incredibly powerful if you want to find transitions between different juggling patterns. All this information can be neatly captured in a graph which is known as a state diagram as in Figure 1. Here, the nodes of the graph are the states and the arrows represents all the possible throws we could make from that state. In this case there is a “height restriction” of not being allowed to throw higher than a 4 to not make the graph too big.
From a given state the arrows indicate to what other states we could travel. The way we get to a different state is by making a throw and the arrow has a number which indicates what kind of throw we need to make to go from one state to another. State diagrams in some sense give an alternative definition of a juggling pattern. That is, a juggling pattern is a closed loop in a state diagram.
Even though we did not explain what states really are it can be insightful to consider an example where we discuss some 2 ball patterns together with Figure 1. You will have to believe me that the strings I give here are in fact juggling patterns, which is something that again is not true for any given string of numbers. The 2 ball pattern 411 can be found in Figure 1 by starting at the state (1100) and then throwing a 4 to arrive at (1001). Then, we throw a 1 to arrive at (1010) and finally we throw another 1 to get back to our initial state of (1100). Juggling 411 is no more than repeating this loop over and over again. Another pattern with 2 balls is 40 which we see in Figure 1 visits the states (1010) and (0101). If we are juggling 411 and want to transition to the pattern 40 we need to find some concatenation of throws which allows us to stop with the loop 411 and go into the loop 40. In this case the patterns 411 and 40 have a state in common, namely (1010). This means that we can happily juggle 411 and at some point decide to not throw the final 1, but instead simply start throwing 40 and keep repeating this loop. It could happen that the pattern I am juggling and the pattern I want to juggle have no states in common. In this case I would need to do some additional throws to help me to travel from one pattern to the other.
All of the topics (and much more) are discussed in a more in depth setting in my bachelor thesis1. However, this thesis is written for a mathematical audience and might be not so inviting to read for someone without a mathematical background. The mathematical formalisations used there are necessary to do rigorous proofs, but not at all required to talk about this topic.
For those interested in more; much of the thesis is based on “The Mathematics of juggling” by Burkard Polster2 which has a, still mathematical, but much less formal tone.