About the Author

Jacob Sharpe grew up in Canton, Massachusetts, and has been juggling and doing circus since the age of 10. For the last four summers he has performed with Circus Smirkus, the award-winning international youth circus, all around New England. He and his brother competed a few years ago at the Festival Mondial du Cirque de Demain, which is one of the two most prestigious circus festivals in the world, performing their world class partner diabolo act. Jacob is class of 2011 at MIT, and plans on majoring in Physics and Electrical Engineering.

The Science of Juggling

by Jacob Sharpe


An astounding number of juggling disciplines offer surprisingly deep insight into a variety of important fields of math and science. The siteswap notation of juggling is a way of creating new patterns using mathematical laws; the diabolo illustrates principals of angular momentum and inertia; balancing objects demonstrates the concepts of center of mass and feedback-control loops; and the learning of complex physical tasks like juggling serves as an interesting platform by which to explore neuromuscular biology. Here we propose an exhibit exploring these concepts, centering on a supervised juggling practice area. We propose to have an inverted pendulum robot and video juggling simulator to illustrate these concepts and attract visitors.

Background and Significance

Juggling has been around since the dawn of civilization; humans have a natural tendency to work obsessively at mastering challenging yet pointless skills. Over the last century, juggling has grown to encompass a huge range of skills involving props of completely different natures. Jugglers are well respected as performers in circus and variety shows, and even command top spots in Las Vegas casinos. But might such a hobby have a place in a science museum? It turns out that a number of juggling disciplines provide a surprising amount of insight into important concepts in physics and mathematics.

In 1985, three people came up with the concept of siteswap notation independently of one another. Mike Day of Cambridge (UK), Bruce Tiemann of Caltech, and Paul Klimek of Santa Cruz devised a way to name some of the more complicated juggling patterns by assigning different throw types numbers, and came up with the same rules for how these numbers interacted.1 The explanation I give here was largely influenced by that given by Colin Wright and Andrew Lipson at www.juggling.org.2

Suppose that a person is juggling three balls to music, such that each throw happens on a beat of the music. Note that each ball is thrown in turn, and the balls are always thrown in the same order. Now suppose that when he juggles higher numbers of balls, he juggles to this same beat, but throws higher to compensate. Consider him juggling four balls. For him to follow this beat—alternating his hands and throwing all the balls the same height—all the throws must go back to the same hand they came from, because every fourth ball is the same ball, and every fourth throw is the same hand. When he's juggling odd numbers, however, the balls cross, because every fifth ball, say, is the same ball, but every fifth throw is made by a different hand. Now he assigns certain throws certain numbers: 3 denotes the throws made in a three-ball cascade, 4 denotes the throws made in a four-ball pattern, 5 denotes the throws made in a five-ball pattern, etc. Given this, a four-ball pattern is just …44444444…, and the analog is true for three and five balls. If he looks more carefully at what's happening in a four-ball pattern, he'll see that each ball is thrown again four beats after it was originally thrown; in 5, the balls are thrown five beats later, and so on. Now comes the tricky part. The juggler is juggling four, and makes a throw like one he would make in five balls. He's done …444445. If he thinks about what will happen if he continues throwing 4s, he'll see that the 5 and the next 4 will collide; five beats after the 5 is the same place as four beats after the throw after the 5. There will also be a gap four beats after 5 where a 4 would've normally gone. To fix this, he could instead throw the next ball to a height so that it would fill the gap, thus avoiding the collision. Since the gap would be four beats after the 5, it's three beats after the throw after the 5; so if he throws a 3, he can continue with more 4s. He's now juggled …44444534444… successfully! Some other patterns that work with four balls are 633, 7333, or 534, where a sequence with no ellipses denotes a repeating pattern. It turns out that all sequences must average out to exactly the number of balls being juggled, or the sequence won't work.

Now all that's left is to deal with the special cases of 0, 1, and 2. In juggling, a 1 denotes a quick pass from hand to hand, so that that object is thrown 1 beat later, as might be expected. A 2 denotes a hold: if the right hand throws an object as a 2, a throw will then be made by the left hand, and that object must be ready to be thrown again by the right hand. Simply holding the object in the right hand accomplishes this. A zero merely means that a hand has nothing to throw on that beat, which doesn't necessarily make the pattern invalid. This notation is called siteswap notation. As discussed below, it has vast potential as a mathematical concept, and is in general interesting to think about.

Another prop that has potential as an example of some of the laws of physics is the diabolo. The diabolo consists of two rubber cups connected by a metal axle, and is spun on a string tied between the tips of two controlling sticks. The diabolo is rolled along the string in one direction so that it picks up speed, and once spinning fast enough, can be manipulated by the handsticks through countless different figures. It illustrates the ideas of inertia and angular momentum, as it won't tilt or change orientation if it isn't touched anywhere but on its axle.

For non-jugglers, most of the feats jugglers routinely perform seem humanly impossible. When people juggle seven balls, how can they possibly be paying attention to all the balls at once, having to see each ball to catch it, while making high accurate throws, all at a high rate of speed?

Another common juggling trick is the balancing of objects on the juggler's hands, chin, nose, forehead, or even foot. But the same concepts are essential for everyday function. When people walk, they are deliberately going off-balance in one direction, then transfering to the other foot. The physics behind this involves center of mass, an interesting concept in physics, and illustrates a feedback-control loop, an important concept in mechanical engineering and robotics.

For non-jugglers, most of the feats jugglers routinely perform seem humanly impossible. When people juggle seven balls, how can they possibly be paying attention to all the balls at once, having to see each ball to catch it, while making high accurate throws, all at a high rate of speed? For jugglers, however, the catches happen naturally; they just pay attention to the top of the pattern, and think about making their throws go right up the center. This is because they've developed extensive muscle memory for how to respond to a falling ball to catch it, and how to throw to a certain height in a certain direction. The neurology behind juggling is very interesting, and makes seemingly impossible tasks more understandable.

Juggling can actually increase the size of the brain, as shown by a study involving jugglers that showed that the adult brain alters its shape for reasons other than simply aging or disease, which had previously been believed to be the case.3 In the study, twenty-four subjects were given a brain scan, then over the next three months, twelve of them learned how to juggle three balls, while the other twelve learned no juggling. A second brain scan was conducted, and after a period of three months of no juggling by any of the subjects, a third brain scan was conducted. The study found that grey matter in the control group didn't change, but that there was a marked increase for the juggling group from the first to second scan, and a decrease from scan two to three, showing both that juggling and other complex fine motor skills can increase the size of the brain, and that the brain will in fact change in shape and function based on how it's used.

As an exhibit, "The Science of Juggling" has a lot to offer museum visitors. It will intrigue them because it's unexpected, and juggling in itself is an interesting subject even to those who can't do it. The mathematical and scientific concepts explored in the exhibit are important ones, and using juggling to teach them will make them seem more fun and interesting. Juggling is also something that many view as a silly hobby, and showing the more technical side of it will hopefully change this misconception. Finally, juggling is just a fun activity, and the opportunity to learn some basics and learn more about it will be fun for visitors.

Proposed Project

The exhibit will center on a sectioned-off area where visitors will have a chance to try their hand at several different juggling props, supervised by staff. Along the half-wall partitioning this section, beginners will find instructions on how to learn the basics with these props. On the walls surrounding this area, detailed descriptions of siteswap notation, balancing and inverted pendulums, inertia and gyroscopes, and juggling neurology will enlighten visitors on the potential for mathematical and scientific depth that juggling has to offer. In one corner, flanked by descriptions and facts about balancing objects and yourself, a robot will balance an inverted pendulum. On another wall, a video screen will simulate a multitude of complex juggling patterns described by numbers through siteswap notation. For a proposed layout of the exhibit, see Figure 1.

The siteswap portion of the exhibit will command the largest amount of wall space, as it is the topic that has the most to offer. The leftmost area of the wall will contain a description of how siteswap works. It will start from a description of the basic four ball and five ball patterns as a string of 4's and a string of 5's respectively, then look at increasingly complex variations in the notation. The section will introduce the concepts in a slow, easy to understand way, making sure an uninformed visitor will be able to follow each step, in much the same way as it is explained in the section above. Each pattern will have accompanying pictures to show what's being described. This area will also teach viewers how to construct their own siteswap patterns.

To the right of this description will be a large video screen showing a siteswap simulator. This will consist of a computer graphics man throwing balls, clubs and rings through a siteswap pattern displayed below him. As each throw is made, the corresponding number will light up, and there will be markers in the background indicating the maximum height of each throw, so viewers can see what number the balls already thrown are assigned. The video will play at a slowed down rate so that viewers can more easily follow the pattern. It will alternate between patterns, choosing randomly from a pre-programmed list of sequences.

On the wall to the right of the screen, panels will discuss the further potential of siteswap. A diagram of the different parabolic paths that objects take for different throws will show the relative heights of differently valued throws. This will also show the energy it takes to juggle different numbers of balls, showing, for example, that a standard seven ball pattern, throwing .2 lb balls at a height of 8 ft., burns calories at a similar rate to running long-distance. In other words, good jugglers have to be in excellent physical shape. This section will also briefly discuss further mathematical potential in siteswap, involving, among other things, group theory, discrete mathematics, and combinatorics. Furthermore, it will discuss potential variations in siteswap, such as using it to describe synchronous patterns, mutliplexes (throwing more than one ball from one hand), and passing. This area will also have some images of various ways to show siteswap, many of which can be visually intriguing in their own right. A siteswap can also be braided, if the right and left sides of the braid denote right and left hands.

The next section visitors will encounter, along the back wall of the exhibit, will concern the physical concepts behind Chinese yo-yos, their cousins the diabolo, and regular yo-yos. It will explain in rudimentary ways the concepts behind gyroscopes, including the idea of angular momentum, and demonstrate these concepts using the classic bike wheel and axle gyroscope. All three of the props mentioned above take advantage of the fact that a spinning object is reluctant to change its axis of rotation, and can be thought of as gyroscopes. A diabolo stays on the string because, while it is spinning, forces that would normally cause it to fall off the string are much weaker than the forces that keep it spinning the same way. This section will also introduce the concept of inertia, and talk about how modern diabolos and yo-yos are engineered to have optimal inertia and minimal friction. This section will focus more on the physical concepts than the juggling props; the props will be used as a means to explain the concepts.

The diabolo section will also include explanations for two seemingly incongruous tricks with the prop, the high throw and the elevator. A diabolo can be thrown to remarkable heights, because pulling apart the sticks can very quickly change the height of the diabolo, and combined with lifting with the arms and jumping, this can result in seemingly impossible high speeds. Many three-diabolo tricks also use siteswap, and a brief description of the difference between juggling siteswap and diabolo siteswap will be included. Diabolo siteswap involves only one hand, so even and odd numbers are no different. A 1 entails a pause with one diabolo in the string, during which a quick trick can be performed, and a 0 entails a pause with nothing on the string, during which a pirouette or jump rope can be done. This section will also explain the common elevator trick, where the diabolo mysteriously defies gravity and climbs the string. This happens because the string has been wrapped around the fast-spinning diabolo, and when the bottom hand pulls down, the friction is great enough that the diabolo will climb the string. This part will include diagrams helping to explain the concepts.

As an exhibit, the Science of Juggling has a lot to offer museum visitors. It will intrigue them because it's unexpected, and juggling in itself is an interesting subject even to those who can't do it.

To the right of this section, in the back right corner of the exhibit, will be the section on balancing. It will discuss the concept of center of mass, explaining how it is located and what it entails. Balance of any object, be it yourself upon a unicycle, or a ladder on your chin, involves keeping the center of mass directly above the point of contact with the ground (or the balancer's chin). A few specific cases will be discussed: balancing a pole on your chin, balancing on a rola-bola, and riding a unicycle. In the specific cases, the exhibit will talk about technique, and how the motions you make serve to move the center of mass of the object with respect to its point of contact with the ground. It will also explain that the longer an object is, the longer it takes to fall over, so larger objects, while they look more impressive, are actually very easy to balance. Objects like acrylic balls, pencils, and hats are, on the other hand, very difficult to balance, requiring extremely fine tuned muscle control. This section will also talk about how we walk and stand still on one or two feet using the same basic concepts.

In the corner will be an inverted pendulum robot. This consists of a long pole (the inverted pendulum) attached at the bottom to a robot that will move along a track to stay under the center of mass of the pole. The pole can swing below as well, and the robot would be capable of swinging it up to an upright position. Visitors will be able to make it do this by pressing a button to start it up, and then can press various other buttons to disturb the equilibrium and watch the robot respond. The problem of how to keep an inverted pendulum upright is a standard dynamics and control problem, and is a simple example of a feedback control system. The panels near the robot will describe to a limited extent the mathematics and physics behind the feedback loop; the particulars will depend on the robot designed for the exhibit, but most use a sensor of some kind to determine the angle with which the pendulum is oriented, and move accordingly. The panels will also describe how the feedback loop works in terms of basic electrical engineering. The robot will help people better grasp the concept of balance, and show that a lot of physical things we do come down to feedback control loops.

The final instructional section will deal with how the brain handles juggling and skills like it. When a person is learning a new juggling pattern, he must focus on the specifics of each individual throw, each of which requires specific muscle movements that may be hard to get exactly right on their own, never mind in conjunction with the movements for many other throws. Similarly, when learning how to balance very small objects, the mind must be able to detect miniscule changes in the weight and position of the object being balanced, and make equally miniscule adjustments in the position of the balancing body part. It is amazing that the brain can develop such an incredible skill given that it was at one point totally incapable of even the most basic versions. This part of the exhibit will discuss how muscle memory, hand eye coordination, and the ability to seemingly pay attention to multiple objects at once are developed through the formation of new neural pathways in the brain. It will also discuss how juggling can increase the size of the brain, and describe the results of the study described above.

In the central area, there will be a receptacle of juggling balls on the left and a receptacle of peacock feathers on the right, along with a receptacle of 2 ft. long balancing sticks. A peacock feather is easy to balance, and will be a fun warm-up for visitors to try their hand at. The balancing sticks will be for those desiring more of a challenge. Along the right side of the half-wall separating the practice area from the rest of the museum, there will be plaques with instructions on how to balance an object on your hand, foot, chin and forehead. On the left side, there will be step by step instructions on how to learn a three ball cascade. This includes the drills that should be learned with one ball and then two balls, as well as tips on how to actually juggle three. Visitors will move down the wall and try different steps until they are ready to move on. Two staff will be working at all times to supervise the central area: one to make sure it doesn't get too crowded and to ensure that no one is stealing props, and another to assist visitors struggling with the skills. On the back area of the half-wall, there will be tips on learning more advanced tricks, such as four balls, five balls, or clubs.

The exhibit will appeal to all types of visitors. Those seeking more mundane entertainment can satisfy themselves watching the siteswap simulator, playing with the robot, or trying their hand at some basic balancing. Those interested in more intellectually challenging material can learn siteswap notation, and gain a deeper understanding of the concepts in physics and neurology the exhibit explores.


The exhibit will be comparatively inexpensive, and as a relatively small scale exhibit, this is to be expected. The inverted pendulum robot makes up a large part of the exhibit's budget, but is nothing extravagant compared to typical exhibit costs. Given the need for staffing, and the potential to lose props, a large cushion fund would be necessary to sustain the exhibit over its first 6-12 months. Table 1 presents the estimated costs:

1. Knutson, Allen. Siteswap FAQ. Juggling Information Service; c1996 [last modified November 10, 1993] [accessed April16, 2008].
2. Wright C, Lipson A. Introduction to Siteswaps. Juggling Information Service. Solipsys Ltd.; c1996 [last modified May 10, 1996] [accessed April 16, 2008].
3. Draganski B, Gaser C, Busch V, Schuierer G, Bogdahn U, May A. Neuroplasticity: Changes in grey matter induced by training. Nature. Nature Publishing Group; c2008 [published January 22, 2004] [accessed April 16, 2008].

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