23 January 2014

Notations, Patterns and New Discoveries

Dr Colin Wright

While ideas are undoubtedly more important than mere notations, the power of a good notation cannot be over-stated. As an example of this, in the mid-1980's a notation was developed for juggling tricks. It was found when using this notation that there were hitherto unexpected connections between existing tricks, and emerging patterns in the notation suggested the existence of new, previously unknown tricks. These in turn led to new ways of thinking about, teaching, and learning existing tricks, as well as providing new material on which to build.

There's a nice little result in mathematics, number theory in fact, that goes like this:
Think of a number
Multiply together all the numbers less than it
If your number was prime, it will divide evenly into the final result.
If your number was not prime, it will not.

Examples:
 • Start with 5 • Start with 6 • Take 1 x 2 x 3 x 4, giving 24 • Take 1 x 2 x 3 x 4 x 5, giving 120 • Add one, resulting in 25 • Add one resulting in 121 • That is a mulitple of 5, and 5 is prime • That is not a multiple of 6, and 6 is not prime
This result has a curious history. It's called Wilson's theorem, and largely the only thing he had to do with it was that he couldn't solve it.

It was first stated by Ibn al-Haytham (c. 1000 AD)[1], and it was almost certainly known to Gottfried von Leibnitz in the 1680's. But it was mentioned by Judge Sir John Wilson to a mathematician friend, who then published in 1770 it as "speculation."

So far so good, but Waring (for it was he) added the prediction that it was unlikely ever to be proven, because there was no known notation for dealing with prime numbers.[2]

And this is where our story picks up, because although he was not the first to prove the result, the famous mathematician Gauss dispatched the problem, saying:

In our opinion truths of this kind should be drawn
from notions rather than notations.

I'm certainly no Gauss, but it is here that my story begins. My experience of doing mathematics is that I play with the notions, trying to write them down, trying to find an elegant way of expressing my thoughts. And when I hit upon the right way to express them, suddenly the thinking becomes easier. To me, notions and notations are not as separate as Gauss might have made them out to be. John Stillwell in his book "Numbers and Geometry,"[3] makes the wry observation that once Gauss had proven the result, he then conveyed the proof with his notation for the so-called "clock arithmetic."

This was crystallised for me some decades ago now when a friend and colleague said:

The act of doing mathematics is that of inventing a language
in which to talk about the problem. Once you have the language,
the problem frequently becomes tractable. [4]

So today I'll illustrate this, giving a specific example which I trust will be both accessible, and outside your experience. I will demonstrate this process of devising a notation, and show how, sometimes, it can then lead to discoveries.

The problem we face in trying to devide a notation for juggling is that it seems such a fiendishly complex activity. Balls, clubs, torches, machetes, chainsaws, all fly in apparent chaos, understood only by those versed in the arcane arts. To make progress at all we must simplify, and simplify radically. The idea is that if we understand a very small, simple version of the problem, we should then be able to leverage that to the next stage, and thus creep up on our goal in stages.

So to start with we juggle only balls. Then we assume from here on that throws happen to a metronomic beat, and that the throws come from the centre of the pattern, and the catches happen just outside shoulder width. Throws and catches are assumed also to occur at about waist height.

We further limit our options by assuming we only throw and catch only one ball per hand at a time, and that the hands alternate.

We're now left with very few options for finding variations in our juggling. Specifically, when we throw a ball, the time it spends in the air is quantised, because it has to come down at one of the prescribed times for catches. Further, once we know when it comes down, that controls how high it goes, and where it comes down. So we describe each throw by a single number - the time it spends in the air. However, since we don't know what proportion of time the hands spend full (or empty), it makes our task easier to think not of the catch, but of the next throw. Now we can see that because throws are separated by a whole number of beats, each ball spends a whole number of beats in its journey from one throw to the next. Each throw can be described entirely by this single number.

The magical thing about this number is that when we're juggling three balls in the standard pattern, each ball is thrown every third time, so the number to describe the throw is a 3. And there's nothing special about 3. Whenever we juggle the standard pattern for n balls, each ball is thrown every nth throw.

The Standard Pattern with 3 balls:
Back in the mid 1980s it was realised [5] that some of the well-known juggling tricks could be described completely just by the appropriate string of numbers to describe the throws. Obviously when juggling the standard three ball pattern we can write ...3333... and for 8 balls we can write ...8888... and so on, but there is a well-known trick with four clubs. Normally juggled with double spins, throw one club high with a triple spin, and the next club low with a single, each club changing hands. Each club drops into the slot vacated by the other, and the pattern then continues as if nothing happened. Much less impressive when done with balls, it is a useful exercise to practise the exact height required for five ball juggling. The high throw will next be thrown five beats later, so is described as a five. The low throw is a three, so we can describe a single instance of this trick as ...444_53_444...

Another well-known four ball trick is make two consecutive throws as if juggling five, pause, and then restart. This can be described as ...444_552_444... (Exercise: why is a momentary hold described as a 2?)

Another variation is to throw all four balls as if juggling five. Of course, after the first four throws we've run out of balls, but if we wait for a beat all the balls come down in order and we can restart our four ball pattern. We write this as ...444_55550_444... It's no surprise that for that moment when we don't have a ball we describe it as a 0, although we shall shortly see that this raises some interesting questions.

Putting it all together ...
Collecting these different tricks and writing them one above the other, putting at the top the uninterrupted four ball fountain, we end up with this:

... 4 4 4 4 4 4 4 ...
... 4 4 4 5 3 4 4 4 ...
... 4 4 4 5 5 2 4 4 4 ...

... 4 4 4 5 5 5 5 0 4 4 4 ...

The pattern was almost impossible to see when we first wrote these down, but leaving the gap makes it unmistakable. The pattern ... 444 5551 444 ... is clearly missing, and based on the sequence, clearly should be a juggling trick.

But it was a trick we didn't know.

From a four ball fountain throw three balls as if juggling a five ball cascade. Now you have one ball left - DON'T THROW IT! Zip that ball across into the otherwise empty hand. Now instead of waiting for a beat, you can carry on immediately.
Do this constantly, and suddenly it feels a lot like five balls. Three out of every four throws is a 5-ball throw, and the pattern is there, in the air, with just a flicker for a missing ball every fourth beat. Superb practice for 5, and enormously easier as it's only four.

An entirely new juggling trick, discovered through mathematics.

A mystery emerges ...
We've shown that some juggling tricks can be described by sequences of numbers, and that by following patterns we can find previously unknown tricks. Not all sequences make valid juggling tricks, but space does not permit investigation of that particular aspect.
Space-time diagram from 5551
There is another question that emerges, however, when we look at the physical reality. We do, after all, have to hold the ball between catching and throwing. In our previous Space-Time diagram we've made the simple assumption that the hands are full for exactly half the time, and we can see that the throws that come back to the same hand are four beats from throw to throw, three beats in the air, giving a hold time of one. The high throws that change hands are fives, and they spend four beats in the air. The zip across is no time in the air, one beat in the hand, and therefore its "Cycle Time" - the time to the next throw - is 1. All this is just as we might expect.

But what about the 0 in 55550? Every other number is the time from throw to throw, and the time in the air is one less. If we follow that pattern, the 0 should give an air-time of -1. We have predicted the time-travel of a juggling ball. How can that possibly work??

If the hands are full for half the time we end up with a ball in the right hand that has come from nowhere, and has nowhere to go.
Clearly we should have the ball go back in time to become itself, just as required.
If we draw a horizontal line on our diagram it's a single instant of time, dividing past above from future below. In a sense it's a photograph, freezing the action and seeing where things are. The diagram here at left has several photographs, each showing where all four balls are.

In each case there's a ball in the hand and balls in the air, always exactly four of them. Which is right and reasonable, as we are
juggling four balls. By the conservation law of juggling equipment we should always have four balls.

But look at the photograph in the diagram on our right. Here we have four balls in the air between the hands, and another ball in the right hand. Clearly there's something strange happening. But wait! There's more! There's also a ball going backwards in time. That must count as a negative ball, to bring our count back to the required four.
It's an anti-ball!
We can think of the "catch" (where the ball comes from the future) as the mutual creation of a ball/anti-ball pair, and the throw back into the past as the mutual annihilation. Thus we have confirmed the view in modern physics that an anti-particle can be thought of as a particle going backwards in time: a positron is an electron going backwards in time, an anti-proton is a proton going backwards in time, etc. More, since a photon is its own anti-particle it doesn't know whether it's coming or going, but since it travels at the speed of light, Einstein tells us time is stopped.
But E=mc2, so where does the energy come from to create a ball/anti-ball pair? Just as there's a quantum uncertainty principle between position and momentum, there's also a quantum uncertainty principle between energy and time. We know exactly when the throws and catches are happening, so we have a very small uncertainty in time and we can borrow from the quantum uncertainty in energy to create a virtual ball/anti-ball pair.

In truth, the anti-ball can be thought of as subtracting a ball from where we expect one, leaving us with an empty hand when our assumptions would normally require a ball.

And in conclusion ...
So where does this leave us? It certainly doesn't end there. Now there are notations for hand movements, timing variations, patterns involving more than one juggler. We have arithmetic methods for determining whether a given sequence can be juggled, and algorithms for producing all possible juggling sequences with any number of balls. Work continues to make these newer notations simpler, cleaner, and more useful. And more than that, the notation continues to be explored as a way of finding new possible variations.

But it's fascinating how the creation of a notation has led to new notions. It's also remarkable how the notation predicted the existence of patterns as yet undiscovered, paralleling the prediction by Dirac in 1928 of the positron, which wasn't confirmed until 1933. That's the problem with the real world, sometimes it's tricky to deal with it.

But perhaps the real bonus is that clear, concise notations allow the effective communication of the notions. The value of a good notation becomes clear as soon as you have to describe a spiral staircase without using your hands, or how to tie a shoelace.

Or how to juggle.

Notions and notations go together, each supporting the other. As so often happens, is the combination of different strengths that leads to the best results.

Footnotes / References
http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Haytham.html
Edward Waring, Mediationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin)
John Stillwell, Numbers and Geometry,
Dr A S Lipson, personal communication
A draft paper for Scientific American is included in "Claude Elwood Shannon Collected Papers," edited by N.J.A. Sloane and A. D. Wyner, New York, IEEE Press, 1993, pages 850-864).
http://www.solipsys.co.uk/new/Juggling.html
http://www.solipsys.co.uk/new/ColinWright.html
http://en.wikipedia.org/wiki/Siteswap
http://www.cecm.sfu.ca/organics/papers/buhler/paper/html/paper.html

The Author
Colin Wright took his B.Sc. at Monash University, Australia, and his Ph.D. at Cambridge University, UK, both in Pure Mathematics. These days he is Director of Research at a company which makes maritime surveillance equipment, is a part-time teaching fellow at Keele University, and still finds time to give presentations all over the world on "Juggling - Theory and Practice," as well as other mathematical topics.