43-2,p32

Figure 4. To find the throw a given ball got, count the dots below the start position. This ball got a 6.

By using this diagram, it is immediately apparent that all odd valued throws must corss 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.

Figure 5. The 3-ball cascade. The path of the black ball has been traced. Every throw is a 3.

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

Figure 6. The path of the black ball in the three shower. The left hand only throws 1's, the right hand only 5's.

By using this diagram (Figure 6), 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.

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 handoff 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 handspeed is actually three times faster than it looks, which would be apparent if you went back to the 3 3... pattern. Then your handspeed really would be three times faster.)

Figure 7-The 5 2 2 thrown once in a cascade. The "trick" throws originate in the triangles and the paths are traced.

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 5 050. (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.