Notaional Formalizations of the Motion of Permutably Controlled Objects

edited November -1 in OK Go
Juggler's World: Vol. 43, No. 2
A Notation for Juggling Tricks
A LOT of Juggling Tricks.
by Bruce Tiemann and Bengt Magnusson

From time to time, jugglers come up with new patterns that look interesting, and are relatively easy to do once you get the idea of them. But often they're hard to explain or describe. There is a way. In this article, a notation will be presented that not only simplifies the description of tricks, but also, because of its mathematical basis, permits an analysis that generates literally an infinite number of tricks.

Readers familiar with a letter by Charlie Simpson, (JW, Winter 86, p. 31) will find little new here as far as the notation is concerned. One difference, though, is that instead of using our notation merely to compile a library of tricks, it can be used to generate all possible tricks within certain constraints. To give credit where it is due, the notation as presented here was independently (and previously) invented by Paul Klimek, with whom we have had helpful discussions.

The notation applies to one juggler with two hands, throwing alternately left-right-left-right in a steady pattern. However, it is easily generalized beyond these constraints to passing patterns with any number of hands, to multiplex patterns where more than one object is thrown or caught at a time, and to in-sync patterns where the hands throw together or in a syncopated rhythm. We welcome those interested to pursue other applications.

The notation here applies to throw heights relative to one another. It is blind to the identity of the objects, applying equally to balls, clubs, rings, or whatever. For the time being, we'll call them balls. It is also blind to "tricks" like backcrosses, Mills's mess, under-the-leg throws or other such things where the throw height (actually the away-from-hand time) is the same as it would have been in the normal cascade.

In other words, these examples aren't tricks in the sense that something is changing from the notation's point of view - they are all just the ordinary cascade. However, tricks such as the shower, the half shower, the chase (three balls in a five pattern) and an infinite number of similar yet distinct tricks exist that can be done without the juggler making any kind of funny throws - just by varying the throw heights between consecutive throws.

In this notation, a trick is represented by a string of numbers. Each number in the string corresponds to one throw, e.g. a string of five numbers represents five consecutive throws. Since the hands are understood in this article to throw alternately, the first, third, fifth, etc. numbers in the string apply to one hand, the rest apply to the other.

The value of the number dictates how high the throw is. It is numerically equal to the number of balls that would be juggled if every throw were that value. For example, a 3 is the kind of throw made every time in a three cascade, a rather low throw across from one hand to the other. A 4 is a somewhat higher throw (higher because the hand speed is understood to be fixed at the three ball speed) that goes to the same hand that threw it. A 5 is a rather high throw that crosses, a 6 is a very high throw that lands in the hand that threw it, etc.

Figures one and two show the odd throws and the even throws up to 10 on the same scale, which is for a six foot tall juggler making 2.5 throws per second out of each hand, and assuming that catches in one hand are exactly coincident with releases made from the other hand. (We feel these are typical or representative values.) These figures can be used to compare against other "juggletoons."

One other thing: the average of all the throw-height numbers in a trick is the same as the number of balls being juggled - which is obvious if all the throws have the same value, but it is true in general. (More about this later.)

Perhaps an analogy would help here. You might think of a hunt-and-peck typist who strictly alternates typing with his left and right index fingers every character. A "string of characters" - a sentence - can be thought of as a notation for how his fingers move. Lots of q's, a's, and z's and he's going to the left all the time, for example. Every other character will be typed by one hand only. Also, note that the spaces are characters so they must be typed. We'll come back to this later too.

Let's consider the basic three ball cascade. Each throw is the same kind as every other. Certainly some go from right-to-left and others go the other way, but they are mirror images, not really different kinds of throws. In this notation, each throw would be called a 3. It does not mean that the throws go up three feet, or that they take three seconds to return, but rather that a total of three throws can be made (including the given one) before the ball is back in a hand, ready to be thrown again. A three ball cascade would be denoted a 3 3 3 3 3... etc. We can simply say it's the 3 pattern, since that's what you repeat to do it. For every throw, you throw a 3. We say this trick has a word length of one, because it's repeat unit is one character long, "3." There is a way to graphically represent this.

In this picture the progress of time is depicted as a downward motion through the diagram, and the two columns of dots represent the throws made by each of the two hands - the left one for the left hand and the right for the right hand. Individual dots represent individual throws, and the alternating aspect of the throws is taken care of by shifting one column half a dot spacing down with respect to the other, so that the path that connects them zig-zags symmetrically as it goes down. (It's similar to Simpson's notation.)

A ball represented by the solid circle is thrown from one of the dots and lands on another below it. How do you find which dot it lands on? Count the dots! A 3 lands on the third dot below it on the zig-zag path, and a 6 lands on the sixth dot below the starting position. (For example, our 6 is Simpson's "5-beat throw," just add one to his number to get ours.)

By using this diagram, it is immediately apparent that all odd valued throws must cross sides, and that the evens never do. Also, it is apparent that even throws are like the throws you do when you juggle n/2 in one hand. (A 6 is a 3-in-one-hand throw for one throw, for example.)

If the juggler does a black ball, a white one and a cube in a cascade, we can follow them on the diagram:

If we follow the black one, we see that it goes from side to side, and that each throw it got was a 3. Now let's follow the shower:

It now gets a 5, then a 1, then a 5, then a 1. The notation for a three ball shower is 5 1 (repeated). This trick has a word length of two. We see two things here. First, that a 1 is what you do in a shower - a quick hand-off from hand to hand that's never really airborne. That's OK. If it's a 1, there doesn't need to be room for anything else. Second, we get an idea why the shower is so hard for many beginning jugglers. The high throws are 5s. For a given rate of throwing, 5s are a lot higher than 3s - not 5/3 as high but more like 4 or 5 times as high! (Assuming gravity; this result is from physics, not intrinsic to the notation.)

Besides the 1, a couple of other throws need to be mentioned. First, there is the 0. It means you aren't doing any (for that throw). Your hand is empty. If you flash all three high, say, and pirouette under them with empty hands, those empty hands are 0s. This is the 5 5 5 0 0. (The word length here is five.) Remember our typist a little while ago? He had to type the spaces. A 0 is the juggler's pause, or space, in this notation.

The other throw that might not be readily apparent is the 2. This is the kind of throw you'd do when you juggle two (in two hands, remember!) - just one in each hand. Now, in principle, you could let go of them, and throw them up a little bit, but since nothing else visits the hand in the meantime, there is no need to, you can just hold on to them and say you're juggling them. Let's try the pirouette again, but this time go around holding one in each hand - "throwing" some 2s. Now we only need to get rid of one ball with a high throw, and take the other two around with us. This is the 5 2 2. (Really it's the 3 3 3 3 3 5 2 2 3 3 3 3 if you do it once in a long run of the cascade, but you can also run the 5 2 2, by doing 5 2 2 5 2 2 5 2 2 ...etc.

This pattern looks like the three cascade, done more slowly, and higher. Then why is the notation different? Because now you are doing three notation throws per real throw, your hand speed is actually three times faster than it looks, which would be apparent if you went back to the 3 3... pattern. Then your hand speed really would be three times faster.)

As far as the notation goes, we're done. That's all there is to it. We can now deal with any pattern that fits the conditions. Three in one hand is the 6 0. (Remember: two hands!) The six object half shower is the 7 5, the four club towers is the 6 3 3 (and the 6s are thrown with four spins, the 3s with single spins), the five ball shower is the 9 1, and the three ball chase is the 5 5 0 5 0. (You may notice that this last one resembles the three high pirouette example given earlier. However, the order of the numbers is important, and even though only one pair of throws has been switched, these two tricks look completely different.)

Seeing all these tricks enumerated, you might get the idea that you can just string together numbers at random, and behold! a new trick.

You can't. Two conditions need to be met. First, one thing in a hand at a time. (Relaxing this condition leads to multiplex tricks.) Consider the 4 3. The 4 will land four dots later. The 3 will land only three later, but it is the very next throw, and they will land on the same dot. We call this a collision, and something to be avoided!

Second, the average of all the numbers in a trick must be a whole number. This is the number of balls being juggled. If it isn't a whole number, the trick isn't possible. (Don't overlook the 2s and 0s when you figure the average, though.)

In the 4 2 the order of the balls is reversed from what they would have been if they got a 3 3 instead. One ball is delayed by one throw, the other advanced by one to make up for it, and the balls land in permuted positions. If the juggler is thought of as juggling imaginary sites where the balls could be, as he does these tricks, which ball lands in which site is different from trick to trick.

For this reason, we call these tricks "site-swaps." The number of sites is equal to the word length, and may be different from the number of balls actually being juggled.

Assigning a computer to find all possible tricks is straightforward, as the tricks are closely related to the concept of permutations. The number of different tricks at a given word length is roughly proportional to the factorial of the word length. (The factorial function starts small and grows really fast. For example, there is only one "trick" at word length one but thousands at word length six.) There is no need to stop at 491. Since there is no limit to the word length, there is no end to the list of tricks. We now give a list of some tricks you can try.

3 balls: 4 2; 4 4 1 (This is probably the most elementary trick that is not widely known. When run, i.e. 4 4 1 4 4 1 4 4 1, it resembles the box pattern but yet is quite different. Note that the 1 throws go left and right, though with a different ball each time.); 4 4 4 0; 5 3 1; 4 5 1 4 1; 6 3 1 6 1 3 1

4 balls: 5 3 (the half shower); 5 5 2; 5 5 5 1; 5 5 5 5 0; (Notice a pattern in these four tricks?) Also try 6 3 3; 6 4 5 1; 5 6 4 1; 6 6 3 1; 7 5 3 1; 7 3 4 5 1; and 5 6 6 1 5 1. This last one made a big show at the convention and was was used by three competitors. 5 5 6 1 3 is interesting because you can run 5 6 1 as many times as you like before the 3 ends the trick and returns you to the fountain. That is, 5 5 6 1 5 6 1 5 6 1 3 is also a valid trick.

Klimek calls 5 6 1 an excited state trick: it can't be done directly from the fountain, you throw a single 5 first. To start it cold, start with three in one hand and one in the other, and throw the first one from the hand with three in it. Another excited state trick is 6 6 1 6 1. Can you see how to "get into" it from the fountain?

The diagram notation may help. 5 balls: 6 4; 6 6 3 (3s are hard. They're so low you need to look down to see them, unlike 1s which are lower but can be done blind.) ; 6 6 6 2; 6 6 6 6 1; 6 6 6 6 6 0 (seem familiar?); 7 4 4; 7 7 3 3; 7 5 7 5 1; 7 7 7 3 1; 7 5 6 2; 8 5 5 2; 8 4 4 4; 8 8 4 4 1; 8 8 5 3 1; 9 5 5 5 1; 9 7 5 3 1; 10 5 5 5 5 0. For the daring, 6 6 6 7 1 7 7 7 1 6 1 will provide hours of amusement. Some excited state tricks are 7 7 1 (a 6 6 or a 7 5 will get you started), 7 5 7 1 (the poor man's six ball half shower), and 7 7 7 1 7 1.

If you're ready for tricks with six or more, work them out yourself, or send for the computer program. (It has been posted on the juggler's listserver.) You might also want to try the above as four or five club tricks. At the 1990 LA convention, Jason Garfield tried to do a head roll (head spin? What are these called, anyway?) with five clubs. Since a head roll is a 3, he needed a five object pattern with a 3 in it. After a boast and several misses on his own, (he tried the 7 7 7 3 1 without our prompting) we told him about the 6 6 3. He got it on the second or third try, and then did a neck roll, a single spin butterfly throw, and a chin sweep each on the first try.

In summary, we have proposed a notation for juggling tricks that involve differing throw heights. It simplifies the description of these patterns. Also, if you're looking for new directions in your juggling, don't get cross at the occult! Look to the numbers instead, as the mathematical basis for this system leads to a large class of new tricks at all levels of difficulty.
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