Wednesday, January 23, 2013

Spiralling into control - A Program to Unwrap Ulam's Spiral

After a difficult study term, I had the privilege during the holidays to see plans from September come to fruition, thanks to the ingenuinty of one of my colleagues. Fascination with Ulam's Spiral, which launched my interest in exploring the properties of prime-generating polynomials and expansions of the Riemann zeta function, has taken me into the realm of computing.

Vision has become reality, as I now have all the class files correctly compiled in my computer and can, at any time, run the program. What I get is a GUI display of tiles that fill with numbers, as per the winding of Ulam's Spiral, with the added feature of black squares, representing tiles that the number sequence skips.

Playing with the spiral program is a bit like playing Angry Birds, if I can borrow my advisor's analogy. It lots of fun, and at least there's a chance that, by playing with it, you might find something useful. So now I'm playing a game I've dubbed "Prime Chasing", something that is a good way of giving my brain a break from editing.

That is just a game, of course. The real work is much more involved. Presently, my colleague is preoccupied with other work, but we have devised two stages of development of our program to systematically explore ranges of black block permutations and to relate these to the lengths of diagonal lines of primes. The algorithm is simple, but the coding takes a bit of time, a thing which neither of us has a the present.

This project is exciting for me as a student who has a deep interest in prime numbers and their properties. Taking these next two steps will allow me to explore some ideas I have tried to wade through algebraically, to no avail. It is my hypothesis that Ulam's Spiral, as a basis for a larger set of spirals formed by the permutations of skipping blocks, holds some deep answers about the pattern of the primes, and, possibly, to the Riemann Hypothesis. Though the mathematical scope of such analysis is far beyond me, it is my hope that generating such results and refining our project based on the initial set will provide data that will be useful to the community of mathematicians who endeavour to expand our understanding of what numbers are.

At the very least, I hope it is better than a game of Angry Birds.

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