Tuesday, August 25, 2009

My Obsession With Recursive Thinking

Once upon a time, Benoit Mandelbrot (he of fractal fame) wrote a paper that examined a simple question, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension".

Of course, Mandelbrot had a budding theory behind his question. A delicious theory that would eventually lead to his glorious work on fractals.

You see, no matter what figure is given for the length of Britain's coastline, it is not a definitive measurement, for, as Mandelbrot examines in his paper, the measured length of a stretch of coastline depends on the scale of measurement.

Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If you were to measure a stretch of coastline with a yardstick, you would get a shorter result than if you were to measure the same stretch with a small ruler. This is because you would be laying the ruler along a more curvilinear route than that followed by the yardstick. The empirical evidence suggests a rule which, if extrapolated, shows that the measured length increases without limit as the measurement scale decreases towards zero. All of that last bit comes directly from the Wiki entry on Mandelbrot's paper, btw.
Step 4 (of 14) fractal

Assuming your head aches when faced with words/concepts such as "curvilinear" and "empirical evidence" or even "extrapolated," consider this simple exercise. Using a map of the island of Britain, draw the simplest two-dimensional shape possible, a triangle, which surrounds the shape of the island as closely as possible, thus approximating the coastline measurement.

Obviously, this crude triangular shape is highly inaccurate. If you were next to carefully draw all the ins and outs of the coastline on your map, you'd have a different measurement entirely than the first, but equally inaccurate, as it's possible to continue getting closer and closer with your scrutiny of the coastline, down to drawing individual pebbles and even grains of sand. And even that degree of granularity would be inaccurate, for more levels of zoom are possible. There is no point at which one can say that a shape resolutely defines the coastline of Britain. After all, exactly circumscribing the coast of Britain would entail encircling every rock, every tide pool, every pebble which happens to lie on the edge of Britain. And let's not start a discussion about how to define "edge."
Some insist that Mandelbrot said the coastline of Britain is infinite in his paper, but he didn't. What he DID say was that it exhibited self-similarity over a wide range of measurement scales. Infinite length woud require a self-similarity over all measurement scales, which is impossible because matter is quantized.

The coastline of Britain is really being used as a metaphor; genuine fractals can't exist in the real world, because the real world is not infinitely divisible (as far as we know). Eventually you get down to the point where you are measuring around individual atoms; if you go farther, you have to go around each proton, etc., and we can't postulate the precise surface of protons, etc., so the recursion becomes impossible for our purposes at that point. Mandelbrot is actually talking about a fractal in a Euclidean space, where actual points exist and we can have line segments as small as we like. Basically, mathematical shapes are just not limited by the quantum nature of reality.

Mandelbrot also discusses the Koch Snowflake in his coastline paper, but that's a discussion for another time. To start your creative juices flowing, though, here's an animation of it:
All this fascinates me, for keeping the seemingly infinite recursive nature of everything around me in mind, it's a very rapid and intuitive leap to realise I cannot possible understand the truth of ANYthing around me, and that the very idea of "truth" or "normal" are absolute rubbish. And that, dear readers, has given ME seemingly infinite patience and humour to draw on in times of trouble. I'm far less patient with myself, for my perception of same is so much more accurate than of anyone or thing else, but sometimes I can even step back and joyously perceive the beautiful chaos that is Me.

I'm rambling a bit, I know, but I find it very hard, when examining recursion or chaos theory, to focus on a single point long enough to find the simplicity therein. Which, of course, is why I so revere people such as Mandelbrot. Building on other peoples work, he was able to focus on a single point long enough to grab the thread and follow it into the Rabbit Hole. Very, very cool in my book.

BTW, if you're interested in a visual understanding of the Mandelbrot fractal sequence, the full set can viewed here.Step 14 (of 14) fractal

No matter what you believe or value, isn't Life an amazing wonderland? Now get out there and see all those recursive physical and behavioural patterns in your world!


melanie joy said...

Love the fracals! i have a suggestion: your next blog is to be about the only constant in the universe.

spare some Change?

Els said...

Great quantum thought, MJ! You've got me laughing after midnight here.

We ARE the change they told us about.