Number theory and neuroscience are probably not the first things that come to mind when you contemplate the circus. Yet along with the clowns and the fire-eaters, there’s a place for science under the big top. Performers and their coaches have to know, either instinctively or consciously, what the laws of physics and biology will permit them to do; moreover, a surprisingly large number of scientists are themselves circus fanatics, and a few are even performers. Perhaps it’s because the circus proves that science doesn’t stop at the lab door.
The crowds are always awed when the acrobats with Montreal’s Cirque du Soleil start twisting their bodies as they flip. But actually such twisting is, in a purely physical sense, easy. Any object resists efforts to turn it--this is the quality physicists call rotational inertia. Rotational inertia increases as the mass of an object is spread farther from its center of rotation: this seems obvious if you consider that a rotating object (or body) with long extremities has to sweep out a bigger circle than one that is more compact. Covering that extra ground requires more energy.
A performer can describe a circle by rotating around either a horizontal axis running through his hips (he can flip) or a vertical axis running from his head to his feet (he can twist). But his body is centered around those two axes differently. From the axis through his hips, his body might stretch three feet, giving him a high rotational inertia. From his vertical axis, however, his body may extend only a few inches, imparting a low rotational inertia. Thus, in a sense, his body wants to twist more than it wants to flip. You can see this by taking a normally proportioned book and flipping it in the air--it’s almost impossible to prevent it from twisting. What makes twisting such a hard skill, in fact, is that it is too easy: if a performer can’t control it, his landing will be ugly.
Coaches know their acrobats can’t calculate Newton’s laws of motion as they do acts like the back flip shown here off a bar held by two other performers. Yet they have to help their performers take advantage of physics (rather than be taken advantage of). Some resort to white lies. For a back flip, for example, some coaches tell their performers to jump straight up and pull their knees to their chests. There’s no way they actually could. For starters, if they jump up in a truly straight line, they have no angular velocity. Instead of going into a flip, they will simply come straight back down.
In fact, most performers learn unconsciously to lean back a little as they jump. That puts their center of gravity a little out of line with their feet, and the force of the jump can turn their body instead of simply pushing it straight up. When the acrobat then starts bringing his legs to his chest, he’s actually just folding his body, bringing his chest to his legs at the same time. Explains Boris Verkhovsky, head coach of Cirque du Soleil, Overall, the body has continued rotating. So it looks like the shoulders have stopped and the legs have accelerated, since they’re going in the direction of the flip. That’s why so many people get caught in the illusion and start believing it. Verkhovsky himself trains his acrobats to lean comfortably back. But he spares his performers the vector diagrams. It’s not something you directly tell; it’s not an instruction. You’re looking for a system of habits.
In the early 1970s, Mikhail Tsaytin, a sports scientist at the Belorussian Institute of Physical Culture and Sport in Minsk, performed an experiment. He had one performer stand on another’s shoulders, then turned out the lights. They kept their position in the dark. Then he turned on the lights, brought in another acrobat, and had them form a three-man tower. Now when he turned out the lights, the tower collapsed.
To a neuroscientist, Tsaytin’s results are not surprising, since a human tower pushes the sense of balance to its limits. The problem arises because balance (or more generally, your sense of motion) is the product of at least three kinds of signals coming from distinct parts of the body.
The first sensor is your vestibular system, a labyrinth of tubes and sacs in your inner ear that is sensitive to quick, small changes in the position of your head. When your head tilts or rotates, tiny, sensitive hairs in the ear are nudged and trigger impulses to a part of the brain stem called the vestibular nuclei. This region converts the signals into a representation of your head’s movement in space; it helps your body keep itself righted and lets your eyes track objects as your body moves.
Signals also come from nerves throughout your body that are sensitive to pressure and stretching. They register, for example, an ankle joint that suddenly sways or a back that gets pressed into a car seat. Finally, the retinas of the eyes also relay some information to the vestibular nuclei.
Under normal conditions, any one of these systems can give you enough information to sense movement. (That’s why you can have the weird sensation of moving in a motionless train when the train on the next track is moving.) But each system works best at its own frequency. While the vestibular system can pick up the fast, little changes needed for eye control, the touch nerves work on slower adjustments, such as the shifts of weight you make while standing. They complement each other--you want one system to be good where the other one is bad, says David Solomon, a Johns Hopkins neuroscientist. (To get a sense of the different frequencies involved, move this magazine in front of your face fast enough that it blurs. Now stop the magazine and move your head back and forth at the same speed. This time you can read the words because when your head moves, your fast-working vestibular system is doing the tracking.)
Circus performers are always swaying slightly as they try to keep their tower aloft. If their swaying moves any of their centers of gravity too far from their base, they all fall. Yet this swaying is too slow to be detected by the vestibular system. The stretch-sensitive nerves in the ankles work, but at a disadvantage: if someone sways underneath you and you sway with him, your ankles don’t bend and you don’t know you’re in trouble. Tsaytin’s experiment suggests that with two performers there’s just enough information available to stay aloft when the lights are out. But three performers introduce so much movement that they absolutely require sight, which can let them see the world shifting slowly around them--and before it starts shifting very fast.
For centuries mathematicians have had a special fondness for juggling (the tenth-century mathematician Abu Sahl liked to juggle glass bottles in a Baghdad marketplace), but in the past decade they’ve taken their affection to a new level--they’ve invented juggling math. One of the founders of this fusion is Ron Graham (shown here), a mathematician at AT&T; Bell Labs and a former president of the International Jugglers Association.
Any juggling routine is made up of a repeating pattern of hand- to-hand tosses, each of which may take a different amount of time. Graham and his colleagues represent such a pattern with a string of numbers that indicate how long a ball is in the air: a quick pass from one hand to the other is a 1, a high arc may be a 5. Thus, a three-ball shower--in which a juggler passes each ball from hand to hand (1) and then throws it high in the air (5) so that the balls seem to be tracing out a big circle--is 15.
But juggling math is more than notation. Graham discovered that given some simple assumptions--such as, you can’t catch two balls with one hand at the same time--he could solve equations that revealed things about both juggling and mathematics. The average of the numbers in any sequence, for example, equals the number of balls needed to perform the corresponding routine. Not all sequences represent performable juggling routines, and with his equations, Graham can test whether they are legitimate or not. (By punching the numbers into a computer, he can even see what a routine will look like.)
Juggling math, Graham has found, is full of unexpectedly elegant laws. For instance, how many different sequences are there that consist of four throws with fewer than five balls? The answer is simply 54, or 625. That’s a really nice number, says Graham. The question is, what is going on? Often, with a really nice answer, you suspect that something nice is going on more broadly. Graham is now trying to find general laws for such sequences. It’s all fitting into a general matrix, like a periodic table, he says. These things are finding their place.
When we watch contortionists like Nomin Tsevendorj and Ulziibayar Chimed, the two 12-year-old Mongolian performers with Cirque du Soleil shown here, it’s hard to think of them as being different only in degree from the vast stiff majority of us. But physiologically, we’re all sitting on the same spectrum.
Our skeletons need to be lashed together tightly enough so they don’t collapse in a heap, yet not so tightly that they can’t take a step. Part of the solution lies in our muscles. Muscles consist of bundles of fibers that can contract and stretch as the filaments inside them slide over one another. But nerves monitor how stretched these fibers get, and if they reach a certain threshold, the brain commands the muscle to contract. Thus muscles refuse to extend so far that they might damage themselves or a joint. It’s possible, though, to retrain this reflex by extending muscles and holding them there; gradually nerves get more accustomed to a longer muscle. This is how stretching makes you more flexible.
Yet stretching for centuries won’t make most people contortionists, thanks to our body’s other solution to its dilemma: collagen. This protein is the main ingredient of ligaments, which connect bone to bone, and tendons, which connect bone to muscle. Cells scattered throughout this connective tissue produce collagen by twisting together three helices of protein and organizing them into long fibers. When these fibers are relaxed, they are loosely crimped up, accordion-style, and connected to each other by cross-links. When you pull on the fibers (say, by bending your knee), they unfold and stretch. The cross-links don’t take kindly to stretching, however, and once they feel the force, they resist it. When you relax your knee, the collagen fibers crimp up again. There are many variations on the basic collagen recipe, and the body uses them all: some cells produce collagen fibers that are rather stiff, while others make collagen that can stretch quite far. By varying the proportions of each in different parts of the body, our ligaments and tendons can hold us together yet allow us elasticity where needed.
Each of us is born with genes that tell our collagen-making cells which recipe to use. Some of us, however, thanks to an abnormal gene sequence, have cells that produce particularly elastic forms of collagen. These people consequently have a wider range of motion than most of us. Among these folks, some produce an extremely elastic collagen, and they are the ones who can become contortionists (it helps if they stretch their muscles from childhood). For a few, though, flexibility can be too much of a good thing. About 1 in 5,000 people is born with Ehlers-Danlos syndrome-- genes produce collagen that is so loose that their joints can pop out spontaneously. In the syndrome’s most serious forms, the tendons and ligaments are the least of a person’s problems. Blood vessels, the bladder, skin, and the intestines all use collagen to stay elastic, and Ehlers- Danlos can make them dangerously pliable.
Fortunately, though, on the spectrum of flexibility, these troubles are at a far end. We see only the patients who come with problems because of dislocating shoulders and things of that sort, but there are a lot of people who have it who are otherwise totally functional, says Leigh Ann Curl, an orthopedic surgeon at Johns Hopkins. There are a lot of people who walk around with Ehlers-Danlos and don’t even know it.