There are objects—such as clouds, mountains, and coastlines—that are extraordinarily hard to model mathematically. The chaos they contain makes traditional mathematics unable to properly deal with them. Fortunately, there is a special branch of mathematics that deals with these matters, whose most representative exponent is the Mandelbrot set fractals, which open a door to a wonderful and unknown world.
When we see a tree, a cloud, or a mountain, we have no doubt that the object in front of us is—indeed—a tree, a cloud, or a mountain. It is strange that this happens, because these objects never repeat themselves. No matter how hard we look, we will never see two that are exactly the same. However, they have certain properties that let us recognize them as such. The set of these common properties matches those of mathematical objects discovered more than a century ago, which are called, in general, fractals.
The History of Fractals
The history of fractals begins with the appearance of the Weierstrass function. At that time the concept of a fractal did not exist, but its graph certainly contains characteristics that make it a member of that club. Later, objects with similar properties were discovered, almost always as mathematical curiosities but with a stricter definition from a geometric point.
These objects can almost always be built from an initial figure, or “seed”, to which a series of simple geometric transformations are applied. When the number of steps is high enough, the figure obtained is what we call a fractal today. In 1904, Helge von Koch defined a curve with properties similar to Weierstrass’s, which he called “the Koch snowflake”. Years later, in 1915, Waclaw Sierpinski constructed his famous triangle and, a year later, his carpet. But all these objects pale in comparison with those discovered in 1975 by the mathematician Benoît Mandelbrot.
The Mandelbrot Set
Mandelbrot was inspired by the work of Pierre Fatou and Gaston Julia, who in the 1920s had already managed to build extremely complex fractals from repeated applications of holomorphic functions. We won’t show you the equations here, but they are not as complex as the graphics might lead you to think.
Today, a mathematical object is said to be a fractal if it meets the following conditions:
- Being irregular enough to be described in traditional geometric terms.
- Having details appreciable at any scale of observation, being self-similar (its parts resemble the whole).
- Having a Hausdorff-Besicovitch dimension greater than its topological dimension.
- Being definable by a simple recursive algorithm.
It is not enough to meet only one or some of these characteristics for an object to be considered a fractal. The straight line, for example, is not considered a fractal, because despite being self-similar, it lacks the rest of the listed characteristics. Today, computers and their computing power have made possible the generation of fractal images with virtually any level of detail and in relatively short times.
Long gone are the days when Fatou or Julia burned their eyebrows over paper trying to draw their fractals. An algorithm (in pseudocode) like the following allows creating Mandelbrot set fractals in two dimensions:
For each pixel on the screen do: { x0 = x co-ordinate of pixel y0 = y co-ordinate of pixel x = 0 y = 0 iteration = 0 max_iteration = 1000 while ( x*x + y*y <= (2*2) and iteration="=" < max_iteration ) { xtemp="x*x" - y*y + x0 y="2*x*y" y0 x="xtemp " 1 } if ( then color="iteration" else plot(x0,y0,color) }</=>
Anyone who has ever written a simple program in BASIC or C can adapt this algorithm in minutes to draw their own fractals. But in more recent times, some programmers have begun generating these figures in three dimensions. The algorithm is much more complicated and beyond the scope of this short article, but the resulting images are of undeniable beauty.
As you can see, mathematics does not have to be cold or boring. Guys like Benoît Mandelbrot or Gaston Julia have turned the not-always-liked numbers into objects so rich and complex that some even consider them works of art. What do you think?
More 3D fractals at Skytopia.
