Learning to juggle 3 balls

Last year my friend and coworker Jari was displaying some juggling tricks and I thought it looked very neat and satisfying. So, after picking up the balls and failing a few times, I asked him if he would teach me. We set a goal of juggling 200 consecutive throws of a 3 ball cascade and away we went. The 200 was set to ensure consistency, as having a solid foundation here makes it easier to move onto more advanced tricks. In the past I have usually skipped the repetition phase of learning something as it becomes monotonous and lacks the sweet reward of exponential improvement. In some ways, this goal acted as a test for me to see what happens if I stick through the practice.

Progress initially was slow but day by day my goal came closer until after 53 days I hit 230 consecutive throws and catches. The throws weren’t perfect but they were all there. Throughout the training there were a couple small things I struggled with that only became visible in the nitty gritty quest for consistency. I found them to be the most interesting parts of the process so will highlight them here for those who target beyond the flash (A single trick completion, in juggling terms).

Throw from the inside catch on the outside

The gif above highlights how throwing in this way creates space for the balls to pass each other. This is necessary for a continous cascade. When juggling you can focus on the circular motion you make with your forearms. One tip from Jari was to fix your elbows either into your body or imagine they can’t move to force this motion.

Initial grip

In the early stages I had trouble getting the third throw (second with the right hand) to a consistent height. Having each throw the same height gives a predictability that helps to reach the higher catch count. It was strange the first and second were great but the third would often fall short. Upon closer inspection, I realised I was throwing the ball from the center of my hand in a pop motion.

I called out to Jari explaining my predicament and asking him what I should do.He was unsure¹, and so he began juggling and as I watched I noticed a small trick. After throwing the first ball he would loosen the second ball in his right hand so it dropped into the position of the first, ready to be thrown.

After this point, I had some dramatic improvement to my early consistency which meant my juggles started on much better ground. It often feels like while juggling you are fighting instability with each bad throw or catch knocking you farther. Having those first throws solid gives you much more leeway for some mishaps down the track.

Eyes at the top of the arc

One trick Jari told me was to not watch the balls moving and instead just focus on the top of the arc. I felt this improved my performance as it focused me on the goal of getting the balls to the same spot.

Keeping the rhythm

When someone is juggling well you can hear it, there’s this pap pap pap. A consistent spacing between throws can be heard and felt. From time to time, I found it helpful to focus on maintaining a rhythm.

Loosen up

Inevitably some throws are not perfect and the wobbling begins. It’s important to be able to adapt to these throws and try to get them back on track. Jari pointed out that I took a very stiff stance and by loosening it I could have greater leeway. I was able to loosen the stance by widening it slightly, bending my knees and freeing my hips. When I threw my 230 throws, there were multiple times I had thought it completely over but with some twists and turns got back on track.

After completing my goal Jari revealed he had been keeping track of my progress over the course of our adventure. I was very excited.

While looking at the results, there’s quite a strong looking exponential growth in catches that’s visible from around day 25. This does line up in some ways with my experience of learning where I felt I was struggling to land even just a few throws and then after a few of the above lessons I started running up the exponential.

When juggling, we can think of the juggler as having a probability that they will drop the ball each time (I think it would be good to have this increase over time to encompass getting tired but good enough to explore). Then, we could ask what was my drop probability as I learnt? Since we only have the maximum of each of my sessions, we will need to make some assumptions to move forward. We’ve assumed each throw is independent with drop probability p allowing us to model the number of throws until a drop using a geometric distribution. Given this distribution, the expected best of N sessions follows:

where $H_N$ is the N-th harmonic number, the sum of the reciprocals of the first N positive integers:

$$H_N=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{N}$$

We need an assumption for the number of sessions N. Based on my memory it was around 15.

$$H_{15}\approx 3.32,\text{so for a particular session’s best run}R$$$$p=1-e^{-3.32/R}$$$$\text{e.g. for my final day/}R=230\Rightarrow p\approx 1.4\%$$

From this we now have we can estimate the drop probability for each day, so let’s take a look.

With the distribution and p we can also compute the expected average and expected minimum throws

This chart displays some of the qualities of the geometric distribution. Primarily it’s skew towards 0 shown by the smaller distance between the average and minimum and it’s long tail shown by the larger distance between the average and maximum. I do think this does not line up exactly with my feelings of juggling. In the early stages, I felt I had many more bad starts with few throws where I would struggle to get stable and then once I was going there was a higher chance they would go longer. If I tracked the data from every trial I might have expected a distribution more like this.

Then, in the later runs I had far less bad starts and more middle of the road runs and the occasional great run. More like this.

Despite this, I do feel the right tail of the distribution is accurate enough for modeling the maximum. So pressing on, another question of interest could be how do more or less tries impact the estimated maximum you will reach given some drop probability?

I found it interesting moving the tries feels like it lets you tap into further along in the distribution very noticeable in the more exponential section where the differences become larger. This feels true, if you told me I could have 5 versus 20 tries I would be quite sure I could get a better maximum in the 20. I do feel some inaccuracy with those higher number of trials as you will likely start hitting problems with physical and mental endurance and perhaps see no further gains.

One final fun thing we can ask is if I have my current maximum and how many tries I’m going to make what’s the chance in the next x sessions I will reach x throws.

I really enjoy this widget as I feel it demonstrates something I felt while learning. While practicing there was a point I knew I was in the range of my 200 throw goal. When I had reached 146 throws the calculator spits out there was a 15% chance in a 15 trial session that I would reach my goal and this feels decently accurate. I think we have quite a good measure of this underlying distribution as we are learning something that follows this pattern of repeated trials that reduce some error rate (Another example I have experienced is with learning a song on piano). I find it a motivating idea to focus on improving the distribution rather than hitting some goal. The maximum reached in some trials could just be a lucky trial. That being said, we do need something to measure against and personal best is easy to check against and feels good to beat. One improvement that could be made to goals I have in the future is hitting a maximum n times to count as it reduces the likelihood of a lucky strike.

After reaching my goal I had my own experience with juggling but was curious about Jari’s so I put together a few questions to ask him.

What do you like about juggling?

I like many aspects of juggling. The first is probably the process itself, of learning tricks. Tricks are like challenging puzzles of mental and physical coordination, where some curious combination of throws merge into beautiful patterns, that go on and on for as long as you have the stamina and the accuracy to keep them going.

The learning experience of these tricks and patterns, has a nice feeling of progress and instant feedback. Every attempt is low-effort in itself - pick up the objects you just dropped and go again. With practice you notice that now and again you get a bit further than your previous best, and these micro-wins feel like victory every time!

The juggling community is great too. I don’t think I’ve ever met an unfriendly fellow juggler (Post interview Jari wanted to clarify his sample size may not be massive here). It’s very supportive at all skill levels. Sharing tricks and ideas, practicing together, coming up with new tricks. It can be very creative.

How would you describe the feeling of juggling?

Described in one word I would say flow. How one throw fits perfectly into the next if you do it right. You might feel a rhythm, a groove, and different patterns have different beats because of their unique structure. Some are more complex than others and I have to really focus on what I’m doing. Other times I can almost shut my brain off and just relax, pure muscle memory takes over, and I focus a lot on posture, breathing and facial expression (yea, that’s a thing).

While practicing you might feel a completely different set of emotions, depending on how well you handle misthrows and mistakes you inevitably are going to make. A feeling of not being in control, or your body not doing what you want. Confusion, frustration, fear(?)… when your objects collide in the air and come crashing down on your head. Patiently you pick them up again, “just one more try…“, because you know you are close, you almost had it. “Trust the process”, to quote a friend of mine.

How would you distinguish between juggling clubs and balls?

Clubs hurt! Not too much fortunately haha, but yes, you have to accept you bonk yourself in the head once in a while when learning. It’s part of the fun, no?

The obvious difference between the two is their shape and size. This makes them unique in how they behave in the air and how you throw them. Balls are small, fast and agile, easier to correct mistakes with. Clubs being larger, take up more space in the air, so in a way you have to be more precise with your throws.

Rotation is a huge part of juggling clubs, so a lot of practice goes into that. Where you hold them matters, and how you throw them matters too: different tricks require different grip and handling. Much less so with balls, you don’t have to worry about many of these things, so they are a lot easier to get started with. They are also less affected by wind which can be preferred if you juggle outdoors depending on your environment.

Clubs are also great for throwing around your body, it’s something you see a lot: behind your back for example. The handle gives you extra reach and allows you to sort of “fling” them, which makes it a lot easier and fun for someone like me that is not very flexible. It’s not impossible with balls though, just a different technique.

There’s a lot more, like how they approach contact juggling and other styles in different ways, but I haven’t delved much into it, so I leave that for someone else to describe. I will mention passing, however. Passing is when two or more jugglers throw their objects to each other, synchronised in a shared pattern. It’s great fun and I highly recommend learning 3 clubs well enough just for passing alone. Clubs are absolutely the go-to equipment in passing: a good throw fits like a glove in your passing-partner’s hand, while the same feels very awkward with balls.

Advice for those wanting to learn to juggle?

Patience

Learning to juggle is building muscle memory. It takes lots of repetition and “sleeping on it” to have your brain save the progress. Set aside a few minutes every day, it really goes a long way.

Break it down.

Patterns and tricks are built from a combination of throws you need to do in succession. Learn these throws one step at a time and build upon the previous step when comfortable. If any part of a pattern is more difficult than the rest, try to work on that part in isolation. Looking at a video might help, slowing it down if possible.

The fundamental pattern to learn is the 3 ball cascade. There are great videos online to explain the pattern, much better than I can do here in text, but for the concept of breaking it down, it might look something like

1. Practice throwing one ball from the right hand to the left. And from the left hand to the right. Both throws should be consistent, at the same height and feel “equal”.

2. Add the second throw: Right-Left-Catch-Catch, Left-Right-Catch-Catch. Again, repeat until comfortable and equal.

3. Add the third ball, almost there, aim for three throws and catches: Righ-Left-Right, Left-Right-Left

4. When the above feels good, comfortable and equal, at a steady rhythm: can you add one more throw to keep the pattern going a bit? Two more? This is where we start…

… counting!

Try to count your catches. It’s a great way to track your progress on most juggling patterns. Beating your current personal best is a great feeling. I find it very motivating. Tip for when you get to higher numbers is to count only catches in the right hand, then double it to get the total.

What’s next?

Once you have the 3 ball cascade down solid, try to learn more tricks. Not only is it a good next challenge, but I believe it will also make your cascade even more consistent. It’s your body learning to adapt to other types of throws, and also enables you to better handle misthrows, making saves, and keeping a pattern going.

Overall I had lots of fun with the experience and now when three things show up I can have some fun and juggle them, thanks Jari. I hope this inspires someone to try it, it’s a hoot! I felt my patience for sticking through something that takes time improved and an increased appreciation for the details you find on the way. When learning, I try to focus more on the distribution and watch what my teachers do carefully when I can’t understand their explanation.

I wanted to mention that some of this practice took place in breaks from programming and I was a surprised by how much it helped me move through problems. I remember reading a comment somewhere about the funny phenomenon where you call someone over to talk through a problem and just as you start talking, you have the solution. They believed it was because you were engaging a different area of your brain and that running the problem through the differently firing system the problem just flowed through. I believe something similar can happen when you get up and juggle.

A final note is that one of my favorite parts of the experience that wasn’t mentioned was the looping inspiration between teacher and student . I found Jari’s juggling very inspiring and I think he found my practice inspiring. I think it’s much easier to maintain this motivation together as opposed to alone. (I found out that in response to the interview, Jari continued his juggling and improved his club backcrosses). I care because you do and juggling art.

Footnotes

1. There were quite a few times where I would ask Jari a question and even though he can perform the action perfectly could not answer my question. I thought this was a nice illustrative example of an expert blind spot. Whilst Jari could do the action it is completely subconscious and automatic, as seems to happen with mastery. I then come along and ask him to be conscious of the whole thing and there’s this funny moment where he realises he doesn’t know and so can’t answer. To resolve these cases we would both pay attention to how Jari would perform the action looking for the answer. ↩︎