All told, it seems likely that bounce juggling may be easier than toss juggling, but very few people specialize in it.
Claude Shannon, Scientific Aspects of Juggling
Claude Shannon, in his paper, “Scientific Aspects of Juggling“, argued that the energy requirements for bounce juggling are significantly lower than that of toss juggling. He suggested that the follow-on benefits of the lower energy requirement would outweigh other disadvantages of bounce juggling, and so, “bounce juggling may be easier than toss juggling.”
Table 1 summarizes some of the toss and bounce juggling world records (at the time of this writing) for odd numbers of balls[1]. For each row, we have highlighted the “superior” record in green.
| # Balls | Toss | Lift Bounce |
|---|---|---|
| 13 | 15 catches | NA |
| 11 | 34 catches | 12 catches |
| 9 | 304 catches[2] | 480 catches[2] |
If bounce juggling were easier than toss juggling, we’d expect all the green highlights to occur under the bounce juggling columns. While this is true for 9-balls, bounce jugglers have only barely managed to flash 11 balls and not even that for 13 balls. Why?!
Could the second half of Shannon’s quote be part of the answer: there are fewer bounce jugglers than toss jugglers? Some possible reasons for this could be:
- Bounce juggling needs relatively specialized equipment: balls with a highly elastic bounce and a hard, level floor[3].
- Bounce juggling practice could also be more frustrating as the balls don’t just drop at the juggler’s feet after a mistake, they bounce away.
It could thus simply be because fewer people attempt bounce juggling that its records are inferior to toss juggling. Maybe the jugglers who have the potential to smash all bounce juggling records just haven’t got around to trying it yet. Or, could it be that bounce juggling is more difficult than toss juggling? In this post, we will explore this possibility. We will mainly concentrate on lift bouncing patterns for the rest of this post and compare them against toss juggling. We will further restrict ourselves to cascade patterns with an odd number of balls. One final clarification before we dive in. All these records are a result of incredible skill and practice and may have been possible only because of the individual brilliance of the juggler. This discussion however assumes that there is no fundamental difference in the abilities and skills of the best toss and bounce jugglers which could explain the disparity in the records for the two styles of juggling. This may be a flawed assumption, but we proceed with it nevertheless.
Small Numbers
Figure 1 shows the default Juggling Lab simulations for 3-ball tossed and bounced cascades.
Notice that the throw rate for the (lift) bounce pattern is significantly slower than the toss juggling throw rate. To understand why, recall from Numbers Juggling that Juggling Lab has modeled the toss juggling throw rates required for juggling different numbers of balls based on observations of actual jugglers. If the bounce juggling throw rates are to be kept the same as the toss juggling throw rates, the flight times[4] for 3-ball and 5-ball bounce (see Table 2) are such that they can only be achieved by a hyperforce solution and a force solution respectively (see Figure 15 in Anatomy of Bounce Juggling and Figure 2 below) without disturbing any other default settings in Juggling Lab.
| Parameter | 3-ball | 5-ball |
|---|---|---|
| Throws per second T | 2.9 | 4.1 |
| Flight Time F (beats) | 1.7 | 3.7 |
| Flight Time (seconds) = F/T | 0.59 | 0.90 |
To be able to juggle a lift bounce pattern for these cascades, Juggling Lab rescales the timing to use slower throw rates of ~1.63 and ~3.54 throws per second respectively for the 3 and 5-ball lift bounce cascades. These correspond to flight times of ~1.04s for both cases which are achievable through lift bounce. Indeed, two possible solutions of vertical speed exist in this region due to the non-monotonic nature of the lift bounce curve of Figure 2. The vertical speeds could be ~1.59m/s and 2.11m/s as opposed to the vertical speed of 2.89m/s required for toss juggling a 3-ball cascade at 2.9 throws per second and 4.41m/s required for toss juggling a 5-ball cascade at 4.1 throws per second. This information is summarzied in Table 3. We can see that for both 3 and 5-balls, lift bounce juggling requires a smaller vertical throw speed, and hence, lesser energy, than toss juggling.
| Number of Balls | Throws per Second | Vertical Throw Speed (m/s) | |||
|---|---|---|---|---|---|
| Toss | Lift | Toss | Lift 1 | Lift 2 | |
| 3 | 2.9 | 1.63 | 2.89 | 1.59 | 2.11 |
| 5 | 4.1 | 3.54 | 4.41 | 1.59 | 2.11 |
Therefore, it sounds reasonable that 3 and 5-ball lift bounce juggling should be easier than toss juggling as there is no parameter that has “worsened” for the bounce juggler. Interestingly though, the endurance records for 3 and 5 ball juggling are also better for toss than for bounce[5]! Maybe lift bounce for smaller numbers is just too slow and too boring to keep up for long durations?!
7-Balls
For a 7-ball cascade with the default Juggling Lab dwell time of 1.3 beats, the flight time would be 7 – 1.3 = 5.7 beats. The default setting for 7-balls in Juggling Lab is 5 throws per second. This means that the flight time in seconds is 5.7/5 = 1.14s. As shown in Figure 2, for this flight time, three possible bounce solutions exist: Force, Lift and Hyperlift. Juggling Lab defaults to Lift bounce as shown in Figure 3.
However, a survey of various videos of 7-balls lift bounced shows something different. In particular, the video for the lift bounce world record for 7-balls (as of this writing) looks more like Figure 4 with the balls being tossed to a significantly lower height from where the hands are.
The differences in Juggling Lab settings between Figure 3 and Figure 4 are listed in Table 4.
| Parameter | Figure 3 (Default) | Figure 4 (WR) |
|---|---|---|
| Throws per second | 5.0 | 6.32 |
| Hand height at throw (m) | 1.0 | 1.12 |
| Hand height at catch (m) | 1.0 | 1.0 |
| Vertical throw speed (m/s) | 3.22 | 0.12 |
| Ball peak height (m) | 1.53 | 1.1207 |
Notice that in the world record scenario, the juggler has “worsened” the throw rate (which has been estimated from the video) as compared to the toss juggling case. Why? The answer may lie in the calculated parameters listed in the last two rows of Table 4. By juggling as in Figure 4, the bounce juggler can significantly lower the vertical throw speed. It seems that the bounce juggler is willing to use a higher throw rate in exchange for this advantage.
Now, the toss juggler too could choose to juggle at 6.32 throws per second instead of 5 and thus lower the vertical throw speed requirement as well as the peak throw height (see Figure 5). But (s)he [instinctively] chooses not to do so.
It would seem that the bounce and toss jugglers are each arriving at different conclusions regarding what makes their pattern easier to juggle. The bounce juggler chooses to operate at a higher throw rate while the toss juggler chooses to operate at a higher vertical throw speed and therefore, toss height.
Note that for the lift bounce juggler, the speed of the ball at the time of catching and throwing is relatively very low. Further, the ball is traveling upwards at the time of the catch and needs to be thrown in the same direction (upward) for the lift throw. Only the horizontal direction of travel needs to be reversed. In contrast, the toss juggler has to catch and throw the balls at a significantly higher speed and the vertical direction of the ball has to be completely reversed between the catch and the throw (i.e., during the dwell time). The lift bounce juggler therefore, may be able to work with a significantly lower dwell time than the toss juggler needs to manipulate the direction and speed of the ball.
Now, for cascade patterns, the siteswap number = flight time + dwell time, in terms of beats. We are using a dwell time of 1.3 beats (Juggling Lab default) for the discussion here. So each ball in the 7-ball cascade will have 5.7 beats of flight time and 1.3 beats of dwell time between two successive throws. If the throw rate is 5 throws per second, then this would mean the time between two successive throws, or in other words, the duration of each beat, is 1/5 = 0.2s. Seven beats elapse between two successive throws of the same ball of which 5.7 beats is the flight time and 1.3 beats is the dwell time. So the flight time is 1.14s and dwell time is 0.26s. One way to reduce dwell time would be to increase the flight time while keeping the throw rate constant. Another way would be to increase the throw rate, thus reducing both flight time and dwell time. Using the same approach as above, for a throw rate of 6.32 throws per second, keeping the same distribution of 5.7 beats of flight time and 1.3 beats of dwell time, we get a flight time of ~0.9s and a dwell time of ~0.2s. It seems that the lift bounce juggler is opting for the latter method to reduce dwell time.
Why does the bounce juggler not choose to keep the throw rate the same and increase the flight time to reduce the dwell time? This may be because increasing the flight time would typically mean increasing the throw’s vertical speed, the only potential exception being if the operating point is within the small region at the left extreme of the lift bounce curve of Figure 2 where the flight time can be increased by reducing the vertical throw speed. Note that even this property of the lift bounce curve does not always hold. For cases like our modeling of the world record 7-ball lift bounce in Table 4, the flight time increases monotonically with throw speed[6].
Another question that may arise is, why doesn’t the bounce juggler simply take the luxury of the extra dwell time available and keep the timings the same as the toss juggler? As can be seen from Figure 3 and Figure 5 (left), the vertical throw speed needed for bounce juggling would still be far less than that for toss juggling[7]. The guess we hazard here is that the juggler wouldn’t know what to do with the extra time and it would be detrimental to the juggling rhythm. For 3- and 5-balls the juggling rhythm is slow enough for this to not really bother the juggler and be able to make the adjustment. But for higher numbers, the juggler possibly feels compelled to use no more than the dwell time that is absolutely necessary to achieve the required change in the direction and speed of the balls.
Large Numbers
Let us now consider 9-ball world record scenarios from Table 1. The Juggling Lab default throw rate for 9 balls is 5.5 throws per second and Figure 6 shows the default toss and bounce simulations from Juggling Lab.
An analysis of the videos available of the world record attempts shows that while the toss juggling throw rate[8] is fairly close to the Juggling Lab default of 5.5 throws per second, the bounce juggler chooses once again to juggle at a significantly higher throw rate of 7.16 throws per second[9] in the world record video referenced in Table 1. Some deviation from the default hand heights when throwing and catching enables the bounce juggler to further minimize the vertical throw speed that needs to be imparted to the balls. In Figure 7 we have shown one such possibility (on the right) with hand height for throwing = 1.3m and hand height for catching = 1.2m in comparison with maintaining the default hand height (on the left) at the increased throw rate of 7.16 per second.
The right hand side animation in Figure 7 seems to be a reasonable imitation of the world record video. So the bounce juggler is tweaking two knobs to make the pattern easier to juggle: throw rate, and hand height above the ground for throws and catches, that allows the juggler to achieve the optimal rhythm needed to juggle the pattern.
The record for a 11-ball lift bounce cascade is just 12 catches, barely more than a “flash”. This does not give enough opportunity to extract the throw rate of the pattern and perform the kind of analysis and modeling that we have done for the other patterns discussed here. For 13-ball lift bounce, there doesn’t seem to be any recorded evidence of even a flash having been achieved. Could it be because the optimal throw rate requirement for the bounce juggler is continuing to increase with the increasing number of balls and has gone beyond human ability?
Concluding Remarks
For small numbers like 3 and 5-balls, the lift bounce juggler requires a lower throw rate and a lower vertical throw speed than the toss juggler. This agrees well with Shannon’s surmise that bounce juggling might be easier than toss juggling. Yet, for some reason, toss juggling records are better than bounce juggling records even for these small numbers. For higher numbers, it appears that the bounce juggler prefers to juggle at a higher throw rate than a toss juggler. This helps the bounce juggler achieve a lower vertical throw speed. But could the energy saved in throwing the balls at a slower speed be getting expended in moving the hands faster to achieve the higher throw rate? Could it be that achieving this higher throw rate ends up making bounce juggling harder than toss juggling for sufficiently large number of objects? Might that be the explanation for why the toss juggling records are better than the bounce juggling records for more than 9 ball juggling?
Footnotes
- Odd and even numbers are juggled in fundamentally different patterns, namely cascades and fountains respectively. To keep things simple, we restrict this discussion to analyzing the cascade patterns only.
- The toss and lift bounce world records for 9 balls are in terms of time: 55s and 67s respectively. The number of throws have been counted from the videos of these record attempts and the count may have a small error.
- Some bounce jugglers carry their own patch of “floor” with them.
- Siteswap number = flight time + dwell time. We assume dwell time = 1.3 beats (Juggling Lab default) for all calculations in this blog.
- See https://juggle.fandom.com/wiki/World_records.
- This is true whenever the height at which the ball is caught is lower than r times the height from which the ball is thrown, where r is the ratio of the height to which a ball will bounce back to the height from which it was dropped with zero vertical speed. For a perfectly elastic bounce, r = 1 and the ball bounces back to the same height from which it was dropped. For this discussion, we have assumed r = 0.9, which is also the Juggling Lab default.
- One way to convince ourselves of this is because we can see that the bounce juggled ball rises to a far lower height than the toss juggled ball.
- 304 catches/55 seconds = 5.53 throws per second.
- 480 catches/67 seconds = 7.16 throws per second.