Notation, Patterns & New Discoveries

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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.

Please note that a number of photographs taken at this event are available to be viewed on the Gresham College Flickr page.

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.     © Dr Colin Wright, 2014

Dr Colin Wright

Dr Colin Wright is co-founder of Solipsys Limited. He acheived a Phd from the University of Cambridge in Combinatorics and Graph Theory, which are areas...

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