posted May 13, 2009 09:25 AM         

I'm in love

the best fun you can have with fractals- go on youtube and search "mandelbrot set zoom"... infinite fun.

posted May 15, 2009 01:25 AM         

Fractals are one of the more recent discoveries in the field of mathematics. Uncovered in the late 1960s by Benoit Mandelbrot, fractals, simply put, are geometric figures that are made up of patterns and repeat themselves at smaller scales infinitely (Hastings 443). Mandelbrot and other mathematicians showed through their mathematics and computer programming that fractals are also common in everyday life, in nature, our bodies, and even in popular culture.

Benoit Mandelbrot contributed greatly to the study of fractals. Prior to his work, investigations into fractals began as early as the late 1800s, with German mathematicians George Cantor and Karl Weierstrass. There are two basic types of fractals, regular (geometric) and random. Regular fractals consist of large and small structures that are exact copies of each other, except in size. One of the more well known regular fractals is the Koch snowflake, which is made up of small triangles added to the sides of larger triangles to an infinite degree (Hastings 43). Random fractals are more apparent in nature as their small scale structures may differ in detail. It was this type of pattern that greatly influenced Mandelbrot, who gave these patterns the name “fractal,” from the Latin word fractus, which means a broken stone with an irregular surface (Hastings 43). In the late 1970s, Mandelbrot began to study an equation that later became known as the Mandelbrot Set, which under computer magnification, reveals an endless succession of repeating patterns (Hastings 43).

To understand the relation of fractals to nature, it is important to understand some of the math and math terminology behind them. Two important properties of fractals are self-similarity and dimension. Self-similarity is the property where a small portion of the object is similar in shape and structure to the shape and structure of the whole object. For example, picture the coastline of any beach from high up. It looks wiggly. However, as you continue to get closer and closer, the wiggles do not smooth out, but rather the closer you get to the coastline it remains wiggly (Liebovitch 8). There are two types of self-similarity, geometrical and statistical. Geometric self-similarity means that the smaller pieces of an object are exact copies of the larger piece (Liebovitch 12). A good example is the Koch snowflake. If you were to zoom in on any portion of the snowflake, it would be an exact copy of the original whole. Thus, usually only mathematically defined objects have geometrical self-similarity. Statistical self-similarity occurs when the statistical properties of the smaller piece can be geometrically similar to the statistical properties of the biggest piece (Liebovitch 12). Therefore, the smaller pieces of objects with statistical self-similarity are not exactly like the larger pieces, but are very close. I will refer to statistical self-similarity throughout the remainder of this essay whenever I refer to self-similarity, as I will be discussing real life objects, which are not perfectly described mathematically. A prime example are the arteries and veins in the retina, which have a branching pattern of the larger arteries that is repeated in the branching patterns of the veins and other smaller vessels (Liebovitch 14).

Fractal dimension is another important property of fractals. Simply put, the fractal dimension tells us the number of new pieces of the fractal we will see when we look at a higher resolution. There are three types of fractal dimensions: self-similarity dimension, which describes how an object fills space; topological dimension, which describes how points inside an object are connected; and embedding dimension, which describes the space containing an object (Liebovitch 47). Throughout this essay I will refer to self-similar dimension when I refer to fractal dimension. The fractal dimension is calculated from d in the equation N = Md, where N is the number of pieces left after an object is divided M times. For example, if we divide the sides of a square into thirds, we are left with 9 total pieces. Hence, 9 = 32, thus the fractal dimension is 2 (Liebovitch 48).

One of the interesting properties of fractals is that they can be used to model patterns in nature more accurately than convention geometry models. A good example of fractals in nature is Brownian motion. It was discovered by the Scottish biologist Robert Brown, as he observed the movements of small particles in liquid in his microscope. He noticed that the particles made small, erratic, and unpredictable movements, which he attributed to physical causes. Einstein later discovered it was due to irregular thermal changes (Lauwerier 112). We can use fractals to help us understand this motion, or more specifically, fractal curves. Mandelbrot describes fractal curves as “curves for which the fractal dimension exceeds the topological dimension 1” (Mandelbrot 31). The path of each of the particles moves in a fractal-like pattern, or if you were to picture a coastline, the particles’ movements resembled the shape of the coastline only in three dimensions. That means, like the coastline example, the particles have self-similarity; thus, if Brown were to have looked at the particles under a microscope that was 100 times more powerful, he would have observed the same thing. Therefore, Brown was really looking at a spatial fractal curve, which became known as Brownian motion (Lauwerier 113).