Math-o-rama

Mark is home sick with some manner of bronchial unpleasantness, but he fought through the fog the other night to call me on the phone and say this: "I was just thinking about your prime number thing. I was wondering if you could model it as a fractal." Since I had, myself, been thinking about prime numbers as the result of iterative functions (which is another way to say 'fractals'), it has suddenly become a good time for a brief plunge into things math-y (except secretly I will be using math as a metaphor for life, because that's just the kind of guy I am).

First of all, about my prime number thing: it's a trivially easy fact to prove that all primes greater than 3 can be expressed in the form 6n +/- 1, where n is an integer. That is, prime numbers appear on either side of multiples of six (5 and 7 around 6, 11 and 13 around 12, 17 and 19 around 18). The converse, obviously, isn't true (not all numbers of the form 6n +/- 1 are prime), nor are there any inferences to be made about, e.g., 6m + 1 being a prime for some m because 6m - 1 is prime (23 is prime, but 25 isn't). I generally state my "prime thing" as, "All the numbers of the form 6n +/- 1 are prime, except for the ones that aren't." There's more to it than that--it involves graphing them in a particular way such that there's a well defined way of drawing lines on the same graph that will run through the 6n +/- 1's that aren't prime. This may or may not sound all ground-breaking and shit, but in fact it is nothing more than a visual representation of Erastothenes' Prime Sieve.

Primes are interesting (um...relatively speaking) because you can predict nearly everything about them except where they actually are. The sequence that starts 2,3,5,7,11,13... is pretty much a random sequence of numbers--they all share a particular property, but there's no mathematical way, given prime P_n, to calculate P_n+1. This problem is considered so unsolveable (not that the solution is hard, but that there's simply no solution) that the famous (again, relatively speaking--Brad Pitt is not sitting at home trying to solve this problem or nothing) unsolved problem about prime numbers isn't about trying to figure out a nice formula for whether a number is prime or not. It's The Hilbert Conjecture, and it only tries to quantify the distribution of primes based on The Riemann Zeta Function (Don't try to understand that last sentence, but you might want to click on the link, because the pictures are cool).

With digital computers we got fractals and chaos mathematics--somewhere along the line between the Greeks and ourselves people started to notice that nature didn't behave geometrically. In the words of Tom Stoppard:

Thomasina: Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God's truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?

Septimus: We do.

Thomasina: Then why do your equations only describe the shapes of manufacture?

Septimus: I do not know.

Thomasina: Armed thus, God could only make a cabinet.

It's one of the entirely reasonable but oft-unexamined tenets of our scientific knowledge that it is all written in mathematical language. There's a certain tautology to this (science = anything you can describe mathematically, all else is philosophy, language, metaphor, etc.), but on the other hand, math has proved to be awfully prescient and adaptive to the needs of scientists and scientific theory over the years. The physicists at the beginning of the 20th century, for instance, were pleasantly surprised to find that Riemannian Geometry (geometry of curved spaces) had existed for a hundred years or so when they discoverd that space itself was also curved. But it's only been in the last fifty years or so that we've really had the ability to, as Thomasina puts it in Arcadia, graph more than x's and y's. You only have to look at a bluebell to see that nature isn't geometric, and look at a Romanescu Cabbage to see evidence that nature does seem to be fractal--ever repeating, but always a little bit different with each iteration, on down to infinity. Fractals and chaos mathematics have fallen out of favor in the last ten years or so, because thusfar they've produced pretty pictures but few useful results. I think, though, this is another one of those cases where math is just waiting for science to catch up.