# Juggling

Juggling is the game of taking objects out of the air and throwing them back into the air again. It is generally most interesting when the number of objects involved exceeds the number of hands. It is pretty good fun to involve more than just two hands. It is common for jugglers to strive to make the number of objects per hand as large as possible and to seek out interesting objects with which to juggle. Personally, I vastly prefer to seek out new patterns in which to juggle the three objects I typically play with (in my two hands).

The familiar juggling pattern called the cascade, with three balls in two hands, has a basic form (the natural juggle most folk find easiest to learn) in which each object

• is thrown by one hand from roughly level (as to left-right motion) with that hand's ear
• rises, travelling towards the other side, until it reaches a point roughly level with the other ear and then
• descends to be caught by the other hand about a head's width further to its side.

I call this the outward cascade because objects travel from near the middle to somewhat further out (on the other side). As each object is half way between going up and coming down, the hand which is about to catch it releases a throw (thereby freeing itself for the imminent catch) and the hand which threw it catches the object which was in mid air when it threw the one I just mentioned as being at the top of its trajectory. A rather fine trick with which to modify this is to make an occasional throw from the outside, over the top to the outside while continuing the pattern of outward throws with the other balls.

## Pattern Juggling

The time-reverse of the outward cascade, which I refer to as the inward cascade, turns out to be my natural cascade, in the sense that it is what came naturally to me when I learned to juggle. This rather surprised the friends who taught me (who are, by the way, very good jugglers). It turns out that the inward cascade is significantly more unstable than the outward one. This was a bit of a problem at first but produced a delightful effect in the end.

Once I'd learned to juggle two objects in one hand, or in the other (I still have trouble doing two in each at the same time) I found that it was very easy to change between the inward cascade and these column juggles. This is because the inward cascade is so unstable: it wants to fall apart into a mess so much that it's quite easy to let it fall apart but control the way it does so in such a way as to trick it into falling out into the column. Compare this to the Wright brothers, who managed flight by having a controlled unstable system instead of having a stable system. Once you've mastered controlling an unstable system, which admittedly takes more effort than controlling the stable one, it's fairly easy to control how it varies (which tends to be pretty hard with a stable system). How that relates to mental states might be worth exploring.

Later, I added the shower – a juggle in which one hand throws objects high while the other catches these and throws them flat, fast and straight back. Between these three, perpetually changing from one to another, I can build some pretty crazy patterns. These tend not to be very regular (they are seldom periodic for long) but I can vary them sufficiently to permit an attempt at dancing to music. I rather enjoy this.

To date, my juggling of clubs is still pretty poor: I can just about sustain the cascade but columns are still unstable and the shower is beyond me. It will come with practice, I'm sure. Meanwhile, I mainly juggle with good stout hand-sized balls. I used to have a set of these, which I refered to as the Chernobyl limes, that phosphoresced. This meant that when they were taken from a light place to a darker one, they would shine. It also means that they shon visibly under ultra-violet light.

This meant that I could see them even in the darkness of a dance-floor, provided there was a dark light nearby: consequently, I was able to juggle my crazy patterns even when the darkness and, far worse, strobe lights were messing up my vision so much that I would hardly have beed able to see ordinary juggling balls. The resulting patterns had a certain amount in common with the way I dance. Thus I've been seen at raves happily standing in a corner (sometimes somewhere less convenient – it depended where I found the dark light) and juggling. I'm not sure whether anyone else could see any relationship between my patterns and the music, but I thought I could. More importantly, I enjoyed doing this and some folk even claimed to enjoy watching it. Sadly I lost those limes somewhere along the way.

I've never heard of anyone else whose natural cascade is inward instead of outward, nor of any other pattern jugglers. On the other hand, I've known lots of folk who have much more technical skill in juggling than I have: I can scarcely do the easy juggle with four objects and my attempts at juggling five are generally short-lived (though they once helped to brighten a dull evening in a customs shed), while several of my friends can juggle silly numbers of things – often of silly things, in fact. Then again, variety is the spice of life: it's doubtless better that I do something so very different from what my friends do, because it means that there's more that we can do between the lot of us. It also has the pleasant effect that each of us can admire the other for doing what we can't, which is generally good for mutual respect.

## My peculiar history

When I was a student, I had some friends who were both extremely bright mathematicians and skilled, enthusiastic jugglers. As their fellow mathematician, I'd had occasion to join their discussions of their attempts to find a classification theorem for juggling patterns; but their efforts had been stymied by a consistent failure to find a proof for any of the plausible conjectures that came to their minds.

Each autumn, new students come to the university; and assorted organisations within the university seek to recruit new members – the jugglers' club was no exception. As my friends were on their way to the free gathering they'd laid on to entice new members, in my fourth year as a student, our paths crossed and they (unaware that I was not entirely sober) persuaded me to join them. Along with assorted new students, I soon enough learned to throw three balls in the air and take each back in hand again soon enough to throw it once more. However, I was looking upwards, holding my hands at roughly the level of the top of my head; and throwing the balls across above me. My friends steered me towards a more orthodox arrangement, with my hands moving roughly in the height-range of my torso and the balls rising and falling at about the level of my eyes. I'm sure that was an improvement.

Once I'd got the hang of this torso-level deal, it began to seem fairly learnable, so I continued practicing. However, my friend Adam Chalcraft (one of the theorem hunters) was puzzled by what I was doing; he watched, bemused, because his instincts told him I was still doing something wrong, but he couldn't see what. Eventually he noticed I was juggling backwards (see above). He tried to teach me the outward cascade, but I found this much harder than the inward one. Like any wise teacher, he left me to develop the skill I had, rather than trying to cramp me into the skill he initially intended to teach me. What comes naturally to me is a trick to him; what comes naturally to him is a trick to me.

That was in early October, I suppose; by New Year's eve (you can be sure I wasn't entirely sober then) I'd learned (with a few hours practice per week, at most) the basic cascade (albeit my inward variant) well enough to routinely continue it for a hundred throws or more. That my memory remembers it thus should be filtered through a suspicion that a few cascades of collective duration less than a hundred shall probably have stuck in my memory less well than one of duration more than a hundred; but it remains that I was getting to the point of being able to sustain a pattern for a good length of time. I was also getting good enough at noticing when things were going wrong that I often noticed the problem early enough to try to end the cascade gracefully, and generally succeeeded – of my cascades' ends, roughly one in two was to catch everything.

At roughly this time, I also began expanding my repertoire. I began trying to juggle (the basic cascade) with clubs; I also began (with balls) to try to juggle new patterns: first, two in one hand (while the other held a third); later the shower. Almost as soon as I had more than one pattern to juggle, I began switching between patterns – albeit, initially, simply as a way to delay failure ! I also began, as I expanded my repertoire of patterns, to find that I could juggle the outward cascade: I found it (and still do find it) much harder than the inward one, but it ceased being impossible.

Once I could mix patterns and switch between them, I again caused Adam and his peers (the theorem-seekers) to pay attention. I rapidly learned to switch between patterns on whim, which created enjoyable dynamics – and (unwittingly) provided the counter-example to several of the candiate theorems that they'd explored. Once they understood that a juggle need not be periodic – which I was vividly illustrating – they stopped looking for theorems that took repetition for granted and rapidly found a result that they could prove. To the best of my knowledge, it's the best result that can be hoped for in a universe which doesn't work like clock-work ;^>

Written by Eddy.