I just finally got my copy of Mandelbrot’s book on fractals. In his discussion of curve fractals (that is, fractals formed from an unbroken line, isomorphic to the interval (0,1)), he describes them in terms of shorelines rather than borders. I’ve got to admit
that his metaphor is better than mine, and I’ll adopt it for this post.

In my last post, I discussed the idea of how a border (or, better, a shoreline) has
a kind of fractal structure. It’s jagged, and the jags themselves have jagged edges, and *those* jags have jagged edges, and so on. Today, I’m going to show a bit of how to
generate curve fractals with that kind of structure.

Continue reading →

Part of what makes fractals so fascinating is that in addition to being beautiful, they also describe real things – they’re genuinely useful and important for helping us to describe and understand the world around us. A great example of this is maps and measurement.

Suppose you want to measure the length of the border between Portugal and Spain. How long is it? You’d think that that’s a straightforward question, wouldn’t you?

It’s not. Spain and Portugal have a natural border, defined by geography. And in Portuguese books, the length of that border has been measured as more than 20% longer than it has in Spanish books. This difference has nothing to do with border conflicts or disagreements about where the border lies. The difference comes from the structure of the border, and way that it gets measured.

Natural structures don’t measure the way that we might like them to. Imagine that you walked the border between Portugal and Spain using a pair of chained flags like they use to mark the down in football – so you’d be measuring the border on 10 yard line segments. You’ll get one measure of the length of the border, we’ll call it L_yards

Now, imagine that you did the same thing, but instead of using 10 yard segments, you used 10 foot segments – that is, segments 1/3 the length. You won’t get the same length; you’ll get a different length, L_feet.

Then do it again, but with a rope 10 inches long. You’ll get a *third* length, L_inches.

L_inches will be greater than L_feet, which will be greater that L_yards.

The problem is that the border isn’t smooth, it isn’t a differentiable curve. As you move to progressively smaller scales, the border features progressively smaller features. At a 10 mile scale, you’ll be looking at features like valleys, rivers, cliffs, etc, and defining the precise border in terms of those. But when you go to the ten-yard scale, you’ll find that the valleys divide into foothills, and the border line should wind between hills. Get down to the ten-foot scale, and you’ll start noticing boulders, jags in the lines, twists in the river. Go down to the 10-inch scale, and you’ll start noticing rocks, jagged shapes. By this point, rivers will have ceased to appear as lines, but they’ll be wide bands, and if you want to find the middle, you’ll need to look at the shapes of the banks, which are irregular and jagged down to the millimeter scale. The diagram above shows a simple example of what I mean – it starts with a real clip taken from a map of the border, and then shows two possible zooms of that showing more detail at smaller scales.

The border is fractal. If you try to measure its dimension, topologically, it’s one-dimension – the line of the border. But if you look at its dimension metrically, and compute its Hausdorff dimension, you’ll find that it’s not 2, but it’s a lot more than 1.

Shapes like this really are fractal. To give you an idea – which of the two photos below is real, and which is generated using a fractal equation?

The most well-known of the fractals is the infamous Mandelbrot set. It’s one of the first things that was really studied as a fractal. It was discovered by Benoit Mandelbrot during his early study of fractals in the context of the complex dynamics of quadratic polynomials the 1980s, and studied in greater detail by Douady and Hubbard in the early to mid-80s.

It’s a beautiful example of what makes fractals so attractive to us: it’s got an extremely simple definition; an incredibly complex structure; and it’s a rich source of amazing, beautiful images. It’s also been glommed onto by an amazing number of woo-meisters, who babble on about how it represents “fractal energies” – “fractal” has become a woo-term almost as prevalent as “quantum”, and every woo-site that babbles about fractals invariably uses an image of the Mandelbrot set. It’s also become a magnet for artists – the beauty of its structure, coming from a simple bit of math captures the interest of quite a lot of folks. Two musical examples are Jonathon Coulton and the post-rock band “Mandelbrot Set”. (If you like post-rock, I definitely recommend checking out MS; and a player for brilliant Mandelbrot set song is embedded below.)

So what is the Mandelbrot set?

Take the set of functions
where for each , is a complex constant. That gives an infinite set of simple functions over the complex numbers. For each possible complex number , you look at the recurrence relation generated by repeatedly applying , starting with :

If doesn’t diverge (escape) towards infinity as gets larger, then the complex number is a member of the Mandelbrot set. That’s it – that simple definition – repeatedly apply for complex numbers – produces the astonishing complexity of the Mandelbrot set.

If we use that definition of the Mandelbrot set, and draw the members of the set in black, we get an image like the one above. That’s nice, but it’s probably not what you expected. We’re all used to the beautiful colored bands and auras around that basic pointy black blob. Those colored regions are not really part of the set.

The way we get the colored bands is by considering *how long* it takes for the points to start to diverge. Each color band is an escape interval – that is, some measure of how many iterations it takes for the repeated application of to diverge. Images like the ones to the right and below are generated using various variants of escape-interval colorings.

My personal favorite rendering of the Mandelbrot set is an image called the Buddhabrot. In the Buddhabrot, what you do is look at values of which *aren’t* in the mandebrot set. For each point before it escapes, plot a point. That gives you the escape path for the value . If you take a large number of escape paths for randomly selected values of , and you plot them so that the brightness of a pixel is determined by the number of escape paths that cross that pixel, you get the Budddhabrot. It’s fascinating because it reveals the structure in a particularly amazing way. If you look at a simple unzoomed image of the madelbrot set, what you see is a spiky black blob; the actually complexity of the structure isn’t obvious until you spend some time looking at it. The Buddhabrot is more obvious – you can see the astonishing complexity much more easily.