While reading Mandelbrot’s text on fractals, I found something that surprised me: a relationship between Shannon’s information theory and fractals. Thinking about it a bit, it’s not really that suprising; in fact, it’s more surprising that I’ve managed to read so much about information theory without encountering the fractal nature of noise in a more than cursory way. But noise in a communication channel is fractal – and relates to one of the earliest pathological fractal sets: Cantor’s set, which Mandelbrot elegantly terms “Cantor’s dust”. Since I find that a wonderfully description, almost poetic way of describing it, I’ll adopt Mandelbrot’s terminology.
Cantor’s dust is a pathological set. It’s caused no small amount of consternation among mathematicians and physicists who find it too strange, too bizarre to accept as anything more than an extreme artifact of logic in the realm of pure math. But it’s a very simple thing. To make it, you start a line segment. Cut it into three identical parts. Erase the middle one. Then repeat the cutting into thirds and erasing the middle on each of the two remaining segments – and then the segments remaining from that, and so on. The diagram below shows a few steps in the construction of the cantor dust.
Why should it be called a dust? Because geometrically, in the limit, it’s got to be a set of completely disconnected points – a scattering of dust across the original line-segment.
It’s so simple – what is pathological about it? Well, it’s clearly 0-dimensional in the pure sense – as I said above, it’s just a collection of points. Topologically, it’s a set of points with empty neighborhoods. And yet – look at it. It’s clearly not zero-dimensional. It’s got a 1-dimensional geometric structure. But naive topology insists that it doesn’t. But there’s worse to it. It’s a similar problem to what we saw in the shape-filling curves. Clearly, the dust disappears into nothingness. Every part of it has zero length – it’s seems like it must converge to something very close to the empty set. And yet it doesn’t: the set of points in the Cantor dust has the same cardinality as the cardinality of the set of points in the original line.
For those of us who came of age as math geeks in the late 20th century, this doesn’t really seem that strange at all. But you’ve got to remember: the 20th century was a time of great turmoil in math. At the beginning of the century, the great work was solving mathematics: turning all of math into a glorious, perfect, clean, rational, elegant edifice. The common belief of the time was that math was beautiful and perfect – that while the real world might be ugly, might have all sorts of imperfections and irrationalities, that those real-world flaws could never touch the realm of pure math: math was, in the words of one famous mathematician “the perfect mind of God”. And then came the crash: the ramifications of Cantor’s set theory, Gödel’s incompleteness, Church and Turing’s uncomputability, fractals, Chaitin’s strange numbers… The edifice collapsed; math was flawed, imperfect, incomplete as anything else in the world. It was hugely traumatic, and there was (and in some circles still is) a great deal of resistance to the idea that so much irregularity or ever irrationality was a part of the world of math – that the part of math that we can really grasp and use is just an infinitessimal part of the monstrous world of what really exists in our abstractions.
But getting back to the point at hand: what does Cantor’s dust have to do with information and noise?
Imagine that you’re listening to sound through a telephone wire with incredibly precise recording equipment. You’re sending a perfectly clear sine-wave over the line, in order to see how much noise there is.
You start pretty high – you only want to record noises that exceed, say, 20% of the amplitude of the basic sine-wave. You wind up with a pattern of bursts of noise. Those noises are scattered around, temporally. Now, mark off every time period of greater that 5 minutes where there is no noise. Those are gaps in the noise – the largest gaps that you’re going to look at. Now look in the bursts of noise – that is, the periods of time where there was no gap in the noise longer than 5 minutes. Look for periods of 1 minute where there wasn’t any noise. In between the 5 minute gaps, you’ll get a collection of smaller 1 minute gaps, separated by smaller bursts of noise. Then look into those 1 minute gaps, for 10 second periods with no noise – and you’ll break the bursts of noise up further, into bursts longer than 10 seconds, but shorter than a minute. Keep doing that, and eventually, you’ll run out of noise. But turn down your noise threshold so that you can hear noise of a smaller amplitude, and you can find more noise, and more gaps, breaking up the bursts.
If you look at the distribution of noise, one thing you’ll notice is that the levels are independent: the length of the longest gaps has no relation to the frequency of smaller gaps between them. And the other thing you’ll notice is that the frequency of gaps is self-similar: the distribution of long gaps relative to sections of the recording of long length are the same as the distribution of short gaps relative to shorter sections of the recording. The noise distribution is fractal! In fact, it’s pretty much a slightly randomized version of Cantor’s dust.
Understanding the structure of noise isn’t just interesting in the abstract: it provides a necessary piece of knowledge, which is used regularly by communication engineers to determine the necessary properties of a communication channel in order to ensure proper transmission and storage of information. Recognizing the fractal nature of noise makes it possible to better predict the properties of that channel, and determine how much information we can safely pump through it, and how much redundancy we need to add to the information to prevent data loss.