Home: Vignettes: Reflection - What is Fractal Brownian Motion?

Reflection - What is Fractal Brownian Motion?

A Mandelbrot set with a touch of extra chaos, seen as a whole.The fractal today is a touch more chaotic than usual. To produce the right effect, I used a formula that added a little fractal Brownian motion noise (fBm) to each iteration. In other words, as each step of the fractal was being drawn, it was subject to a random bump, causing the cloudy effect seen below. Brownian motion seemed very appropriate for the subject, as you’ll soon see.

The fractal is still a Mandelbrot type set—you can even see the familiar studded curves along some of the white edges in the lower right. While this simple variation throws off the whole set, it creates a sort of soft, familiar beauty.

Today’s fractal is adapted from a scene in the story I posted yesterday. If you’ve read it, this should be an easy guess:

The Mandelbrot set, with a little Brownian motion and minor adjustments, has a cloudy appearance up close

Here, we have a similar form, like that which tempted the dragon:

The sun and clouds, seen reflected on a still pool aside Standley Lake

When I took this picture, I was thinking about the various repeating patterns: the still surface of the water, the clumps of vegetation floating on top, the reflected clouds. While thinking about the similarities of these patterns, I considered the unpredictable interaction of living and non-living parts, adding up to the complex beauty captured by my camera. (That is, it is beauty as long as you think lake slime can be pretty—I do.)

In retrospect, I should have considered Brownian motion while looking at the surface. Some bits of debris would move around slightly, yet stayed in the same general spot. It might appear as if the debris was treading water, with intent. Of course, lake goo, as far as we know, is not treading water. (Since this is Standley Lake, The Front Range seen from Standley Lake, near the spot chosen for today's fractalrumored to contain traces of plutonium in the sediment, I suppose any weird mutation is remotely possible.)

Botanist Robert Brown, for whom Brownian motion is named, puzzled over a similar problem. Supposedly, he was watching a bit of pollen dance around in a drop of water, under a microscope. He saw the same thing happen with a dust particle. Brown was pretty certain that dust isn’t alive, but couldn’t explain why it could dance.

In 1905, Albert Einstein, who’d become familiar with movement at the microscopic and macroscopic scales, came up with a simple answer. Click to view a java applet demonstrating Brownian MotionThe relatively large bits of pollen or dust were getting bumped around by tiny molecules of water, which themselves were moving about at random. This observation allowed chemist John Perrin to prove the existence of atoms, ending a debate that stemmed back to ancient Greek philosophers.

For more information about Perrin’s proof and Einstein’s study of Brownian motion, check out this article. For more on fractal Brownian motion in nature, as well as applications in subjects as diverse as flood prediction and biochemistry, see this article.

Fractal images created by the author using ChaosPro. Lake photographs taken by the author. Brownian motion image via Fowler’s Physics Applets.