Wednesday, November 12, 2008
Wednesday Math, Vol. 47: Fractals
Benoit Mandlebrot, a mathematically trained computer programmer who worked at IBM, is considered the father of fractals. His pioneering work The Fractal Geometry of Nature was met with skepticism and outright hostility in the mathematical field. It was more like a picture book than a math text, and exactly what he meant by the word fractal was vague. Vague definitions flew in the face of everything that had been done in math in the 20th Century, and the applications of the processes were also vague, and when not vague, trivial.
My dear friend Mina Millett hated fractals. She suffered from migraines and she said that a fractal was what a migraine looked like. When someone asks me what fractals are, my standard answer is they are the mathematical proof of the existence of paisley.
Snarkiness aside, the definition of fractal processes has been made more precise over time. The important idea in fractals is self-similarity. The major method for the creation of fractals is the iterative process.
The two patterns shown here are simply created fractals. At the top, we take a triangle, cut it into four triangles and remove the middle one. We now have three triangles, so we cut each of them into four parts and remove the middle again. Continue the process over and over. This shape is known as Sierpinski's Gasket, and it has a connection to the much older mathematical object known as Pascal's Triangle.
The second series of squares from a checkerboard pattern is also created by removing shapes from an original pattern, and continuing the process with the smaller shapes that look like the original shape. It was discovered about twenty years ago nearly by accident that an antenna using this shape is extremely effective while covering a small space, and such fractal antennas are the standard in cellphones today.
The idea of self similarity is that an object when magnified looks something like the entire object. In nature, a craggy rock looks something like a mountain, and the branching pattern in a tree or in blood vessels has a similar pattern whether at the smallest branches or the largest.
Jonathan Miller in his book The Body in Question noted the importance of metaphor in medicine, his particular field, and in scientific inquiry in general. Ancient civilizations had no idea what the heart did, for example, and considered it a drum in a person's chest. Until the invention of the pump, there could be no useful metaphor describing the heart's function.
The invention of the computer both opened up what was possible for people to compute and the idea that complex things could be created using a simple set of instructions repeated ad infinitum at both the small and the large scales. When a little randomness was thrown into these processes, things that looked "natural" instead of "mechanical" started to appear. This opened up scientific inquiry into how things are made in nature, that the processes used by a computer to create a fractal that looks like a craggy rock or a leaf on a tree might give us insight into how nature creates craggy rocks and leaves on trees.