“These three words represent a programme of moral beauty. The aesthetics of sport are intangible.”
Pierre de Coubertin, Founder of the International Olympic Committee
“How many can you juggle?”
This would perhaps top the list of Frequently Asked Questions for any juggler. It is well documented[1],[2] how rapidly the degree of difficulty increases as one juggles higher numbers. To manage this increasing level of difficulty, a juggler primarily manipulates two parameters of the juggled pattern. In this blog we will identify these two parameters and try to formulate a mechanism which can predict how a juggler will trade them off against each other for numbers juggling[3].
Citius: Faster
The first parameter is what I’ll call the “throw rate”. This is the number of throws made by a juggler per minute and we’ll measure it in beats per minute (bpm). In theory, a juggler could juggle a greater number of objects simply by increasing the throw rate. Figure 1 shows the Gunswap simulations for 3-ball and 7-ball cascades achieved by using different throw rates.
Figure 1(a): 3-ball cascade at 214bpm (Gunswap default) 
Figure 1(b): 7-ball cascade at 606bpm
The 3-ball cascade simulation uses the Gunswap default setting for beat duration = 0.28 seconds which translates to ~214bpm. Another Gunswap default is 0.8 beats of dwell time which means that for a 3-ball cascade, the flight time is 2.2 beats, i.e., 0.28*2.2 = 0.616s. A 7-ball cascade would similarly have a flight time of 6.2 beats. For this flight time to also be equal to 0.616s, we need a beat duration of 0.28*2.2/6.2 = 0.099s which translates to ~606bpm.
Altius: Higher
The second parameter is the “pattern height”. This is the height of the highest throw above the throwing hand. For cascades and fountains, all throws attain the same height which is also the pattern height. In Figure 1, the pattern height remained constant as we simulated 3 and 7 ball cascades exclusively by changing the throw rate. Figure 2 shows Gunswap simulations where the throw rate is the same for both these cascades but the pattern heights are different.
Figure 2(a): 3-ball cascade at 214bpm (Gunswap default) 
Figure 2(b): 7-ball cascade at 214bpm (Gunswap default)
This is the default behaviour of Gunswap (and most other simulators, including JoePass!). But neither Figure 1, nor Figure 2 reflect reality.
Reality: Citius+Altius
In real life, a juggler combines an increase in both, the throw rate and the pattern height to juggle a higher number of objects. In Optimal Juggling, Jack Kalvan lists averaged measurements taken from videotapes for patterns ranging from 3-balls to 9-balls. Table 1 shows the throw rates[4] and pattern heights used for a 3-ball vs a 7-ball cascade as per Kalvan’s data.
| 3-ball | 7-ball | |
|---|---|---|
| Throw Rate (bpm) | 218 | 286 |
| Pattern Height (m) | 0.33 | 1.8 |
The question I want to explore is, given the throw rate and pattern height used by a juggler for one pattern, can we predict what will be used for another pattern?
Juggling Lab
Interestingly, Juggling Lab captures this behaviour of increasing both throw rate and pattern height to juggle higher numbers. This can be seen in Figure 3 which shows the default simulations thrown up by Juggling Lab for the 3 and 7 ball cascades.
Figure 3(a): 3-ball cascade at 174bpm (Juggling Lab default) 
Figure 3(b): 7-ball cascade at 300bpm (Juggling Lab default)
The throw rates used in Juggling Lab were somewhat different from those suggested by Kalvan’s table. When I asked how Juggling Lab had arrived at the numbers it used, Jack Boyce said, “In a nutshell I timed actual jugglers doing n-ball cascades/fountains to come up with an average bps(n)[5] for a given n.”
Fortius: Stronger
In The Physical Demands of Numbers Juggling (TPDNJ), Jack Boyce explains how and why, in order to go faster and higher for number juggling, the juggler needs to be stronger too.
The underlying factor determining the jugglers’ choice of the throw rate and pattern height combination, could well have something to do with optimization of the strength required to juggle higher numbers. Unfortunately, that will need far more analysis than is within the scope of this blog. Here we will focus only on studying the speed vs height trend of numbers juggling patterns.
The Citius vs. Altius Trade-off
The only relevant formula I was aware of which connected throw rates and pattern heights, was Shannon’s Theorem:
(f+d)/(v+d) = B/H, where
f = flight time,
d = dwell time,
v = vacant time,
B = Number of Balls and
H = Number Hands
Though this formula does not explicitly mention either the throw rate or the pattern height, we can determine them from f, d and v. For the two-handed case we’re concerned with, H = 2 and we can split Shannon’s theorem into two separate equations. With t being the absolute time duration for one beat, we get:
(f+d) = B*t —-> Eq. 1
(v+d) = 2*t —-> Eq. 2
This left me with two equations in four variables: f, d, v and t.
Jack Kalvan, in Optimal Juggling, supplies a third equation: the dwell ratio, r, remains approximately constant across uniform juggling patterns, i.e.,
d/(v+d) = r —-> Eq. 3
We still need one more equation.
Speculation
Let’s take another look at Shannon’s theorem, (f+d)/(v+d) = B/H. In numbers juggling, the RHS increases because the numerator B increases while the denominator H remains constant (=2). According to the data measured by Kalvan, the LHS keeps up with this increase in the RHS by using a combination of an increasing numerator and a decreasing denominator. An obvious idea that suggests itself is that the numerator and denominator follow their respective trends such that their product remains a constant, say k. This gives us our fourth equation:
(f+d)*(v+d) = k —-> Eq. 4
We can now solve these four equations for f, d, v and t in terms of B, r and k to get:
t = √(k/(2*B))
f = k/(2*t) – 2*t*r
d = 2*t*r
v = 2*t*(1-r)
If t is measured in seconds, then there being 60 seconds in one minute, we get:
throw rate in bpm = 60/t —-> Eq. 5
If g be the acceleration due to gravity (= 9.8m/s2 on Earth), then,
pattern height in metres = 0.5*g*(f/2)2 —-> Eq. 6
Note that t, and hence, the throw rate, does not have any dependency on the dwell ratio r. On the other hand, f and hence, the pattern height, depends on both k and r.
An interesting consequence is that if a and b be two different number of objects being juggled, then bpma/bpmb = √(a/b), i.e., the throw rate is proportional to the square root of the number of objects being juggled.
Results
Table 2 compares the results obtained from these speculative formulae against the data collected by Jack Kalvan. I have used k = 0.55 and r = 0.63 to get the “Formula” values indicated in Table 2.
| # of Objects | Throw rate (bpm) | Pattern Height (m) | ||||
|---|---|---|---|---|---|---|
| Formula | Kalvan | % Error | Formula | Kalvan | % Error | |
| 3 | 198 | 218 | -9.2 | 0.34 | 0.33 | 3 |
| 4 | 229 | 226 | 1.3 | 0.63 | 0.65 | -3.1 |
| 5 | 256 | 255 | 1.3 | 0.94 | 1.0 | -6 |
| 6 | 280 | 255 | 9.8 | 1.26 | 1.5 | -16 |
| 7 | 303 | 272 | 11.4 | 1.59 | 1.7 | -6.5 |
| 8 | 324 | 315 | 2.9 | 1.9 | 2.2 | -13.6 |
| 9 | 343 | 363 | -5.5 | 2.2 | 2.2 | 0 |
Under the columns labeled “Kalvan” in Table 2, the data up to 6 balls is measured from Kalvan’s own juggling. For 7, 8 and 9 balls, the data is for Anthony Gatto. Figure 4 plots the bpm vs # of objects as determined from Table 2.
Though the formula generates the correct qualitative trend for bpm, it differs significantly from Kalvan’s data.
Refinements
A closer look at the available data and my formula revealed some details that could have affected the results. These are discussed below.
Kalvan’s Data
I found videos of Anthony Gatto juggling 7 balls, 8 balls and 9 balls on YouTube. By measuring the time elapsed over 100 throws in these videos, I made my own calculations for the bpm. These numbers now matched much better with the formula (see Table 3). Measuring pattern height from the videos is a lot trickier so this parameter has been dropped from Table 3.
Cascades and Fountains
As cascades and fountains are both uniformly juggled patterns, I had been reluctant to treat them differently. Further, Kalvan had found that the dwell ratio seemed to be constant across cascades and fountains, reinforcing my reluctance. However, Jack Boyce convincingly argued that we should expect different throw rate behaviour for fountains and cascades (see TPDNJ). Recall that in our formula, the throw rate depends on k but not r. Thus, I could set k to different values for cascades and fountains while keeping r constant and comply with both Kalvan’s and Boyce’s insights.
Stable, Comfortable Patterns
While tabulating his measurements, Jack Kalvan noted that, “All data was taken from what appeared to be stable, comfortable patterns.” But what is a “stable, comfortable pattern”? For example, the typical juggler will be able to do a “stable, comfortable” 3-ball cascade across a wide range of combinations of throw rates and pattern heights. Presumably, expert jugglers could do the same for higher numbers of objects? Which of these multiple throw-rate-pattern-height combinations had Kalvan measured?
Once again, a passage in TPDNJ struck a chord: “… with 3 cascade and 4 fountain a ball which is thrown too high is easily corrected — just by waiting. In the case of 4 this will mess up the alternating rhythm between the hands… However, with 5 or more this simple means of correcting timing mistakes is not available, one of the reasons five feels qualitatively different from 3 and 4… In effect, the presence of these extra balls coming down forces you to adhere to a relatively strict timing, in the short term.“
Maybe stable, comfortable patterns only made sense for 5 or more balls where the juggler is forced to “adhere to a relatively strict timing”? Maybe even the 4 fountain could be included since we do prefer to juggle in an alternating rhythm. The 3 cascade could however be treated as an anomalous case since there could be many stable, comfortable versions of it.
Skill Level Drift
It is possible that the throw rate of the stable, comfortable pattern might drift over a period of time, as the juggler’s skills develop. For example, Gatto’s 7-ball (2011) and 9-ball (2017) videos that I referred to for my measurements are separated by a period of 6 years. It is quite possible that Gatto would have settled to a different throwing rate for his 7-ball cascade in 2017 than he had in 2011. This is something I have not accounted for even in the refined results that follow.
Refined Results
With throw rates corrected for 7, 8 and 9 balls as per my measurements and other factors (except skill level drift) discussed above taken into account, the new comparison between the formula and Kalvan’s data is presented in Table 3.
| # of Objects | Throw rate (bpm) | ||
|---|---|---|---|
| Formula | Kalvan | % mismatch | |
| 3 | 198 | 218 | -9.2 |
| 4 | 225 | 226 | -0.4 |
| 5 | 256 | 255 | 0.4 |
| 6 | 275 | 255 | 7.8 |
| 7 | 303 | 304 | -0.3 |
| 8 | 318 | 318 | 0 |
| 9 | 343 | 340 | 0.9 |
I retained k = 0.55 for cascades, but used k = 0.57 for fountains in Table 3. Figure 5 shows the comparison between the formula and measured data of Table 3.
If we ignore the 3-cascade for reasons discussed above, then the only major mismatch (~8%) is in the 6-ball case. All other cases match very well.
Extended Checks
I used a couple of other test cases for the formula. The first is the table of throw rates used in Juggling Lab. These are averaged over several jugglers, but still, let’s see how the formula performs against this “average juggler”. Table 4 shows this comparison with k = 0.59 for cascades and k = 0.66 for fountains.
| # of Objects | Throw rate (bpm) | ||
|---|---|---|---|
| Formula | Juggling Lab | % mismatch | |
| 3 | 191 | 174 | 9.8 |
| 4 | 209 | 204 | 2.5 |
| 5 | 247 | 246 | 0.4 |
| 6 | 256 | 255 | 0.4 |
| 7 | 292 | 300 | -2.7 |
| 8 | 295 | 300 | -1.7 |
| 9 | 331 | 330 | 0.3 |
Figure 6 plots the numbers from Table 4.
Again, if we leave out the 3-ball case, we have a pretty good match up to 9 objects, including for 6 objects, unlike in Figure 5.
Beyond 9 objects, the formula predicts an increasing throw rate while Juggling Lab keeps it constant. That led me to the second test case. I measured videos for world records for 10 or more balls and compared them against the formula. I did not have videos of those same jugglers doing a lower number of balls so I had very few points to use for correlation. Table 5 summarizes these results, using the same settings for k as used in the Table 3 comparison against Kalvan’s data.
| # of Objects | Juggler | Throw rate (bpm) | ||
|---|---|---|---|---|
| Measured | Formula | % mismatch | ||
| 10 | Tom Whitfield | 352 | 355 | 0.9 |
| 11 | Alex Barron | 368 | 379 | 3.0 |
| 13 (15 catches) | Alex Barron | 432 | 413 | -4.4 |
The errors in Table 5 are still not too high. Besides, there are a couple of mitigating factors:
- The 13-ball video has only 15 catches which is marginally better than a flash. I don’t mean to downplay the magnitude of Barron’s achievement, but as a data point for the formula, it may not count as a “stable, comfortable pattern”.
- Barron’s juggling may be tuned to a different value of k.
Concluding Remarks
Based on the limited set of data and videos that I studied, the formula for determining throw rates and pattern heights based on the assumption that (f+d)*(v+d) = k seems to give fairly good results. However, this limited correlation is not proof that the formula has correctly identified a principle (if there is any) which a juggler subconsciously adheres to while numbers juggling. But it is interesting that the formula gets as close to real life measurements as it does. Maybe for that reason alone, it is worth incorporating into juggling simulators to enhance the realistic feel of their simulations.
It might also be an interesting exercise to examine this formula further to try and establish if there really is something to it. How and why did the human mind pick on the parameter (f+d)*(v+d) and tried to keep it constant? If, on the other hand, this formula is wrong, then is there something else that determines the throw rate vs pattern height trade off that a juggler feels most comfortable with? For example, Jack Boyce outlined to me how parameters of the circular hand motion that an efficient juggler seems to adopt could be relevant to this problem.
An obvious (and probably unsolvable) challenge in pursuing any such study is the paucity of numbers jugglers in the world who could help generate the required data to compare the model against.
Acknowledgements
Thanks to Jack Boyce for showing me different perspectives from which the problem presented here could be approached. As I have indicated, his feedback directly led to some of the refinements that I made to the formula.
Footnotes
- On page 57 of Siteswap Ben’s Guide to Juggling Patterns, Ben Beever estimates that there are 20 jugglers in the world who can flash 11 balls, a mere 0.4% of the number (5000) who he estimates can flash 7 balls.
- Jack Kalvan in The Human Limits estimates that, “… eventually someone will juggle 13 balls, and flashing 15 doesn’t seem too unlikely.” The current world record for toss juggling is 11 balls by Alex Barron.
- We will consider only two-handed cascade (for odd number of objects) and fountain (for even number of objects) patterns.
- Kalvan did not explicitly mention the throw rates in his data but they can be calculated from the parameter “tau“, which he did list.
- “bps” stands for beats per second = bpm/60.