## Thursday, February 03, 2011

### Over my head in Cantor dust

This is the sort of thing I've been trying to get my head around for my new novel. Below is information from Wikipedia on the Cantor set, which is also, rather more poetically, called the "Cantor Dust." (It may only be "dust" when it's in 3D.)

The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third (13, 23) from the interval [0, 1], leaving two line segments: [0, 13] ∪ [23, 1]. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, 19] ∪ [29, 13] ∪ [23, 79] ∪ [89, 1]. This process is continued ad infinitum, where the nth set is

$\frac{C_{n-1}}{3} \cup \left(\frac{2}{3}+\frac{C_{n-1}}{3}\right).$

The Cantor ternary set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process.

The first six steps of this process are illustrated below.

An explicit formula for the Cantor set is

$C=[0,1] \setminus \bigcup_{m=1}^\infty \bigcup_{k=0}^{3^{m-1}-1} \left(\frac{3k+1}{3^m},\frac{3k+2}{3^m}\right).$

The proof of the formula above is done by the idea of self-similarity transformations and can be found in detail.[7][8]

Yeah, no, I can't read these equations at all. But the basic idea is that the set, as James Gleick explains in Chaos, "the points that remain are infinitely many, but their total length is infinitely small." That concept is the basis for fractal geometry; it's the infinitely large AND infinitely small concept that I am really interested in. Just look at the picture! It's amazing! The universe we live in!

But this is all just to say, I am in way over my head here. At some point I am going to need a real mathematician and/or physicist to read this thing. Also I'll need a psychiatrist, although you probably guessed that already. If you know anyone who'd like to help (probably a few months down the road), please email.