Chaos and Fractals The Mandelbrot Set

Fractals

In his book, Fractals: Images of Chaos, Hans Lauwerier defines a fractal as “a geometrical figure that consists of an identical motif repeating itself on an ever-reduced scale” (Lauwerier 1).  This simple process (called iteration) exhibits very complicated behaviour.  The process is a kind of feedback and produces what in science is called a dynamic law.

For an example, take the following link to a website that gives an animated illustration of the “Koch Curve,” which is formed as follows: a line is divided into three equal parts, on the middle third an equilateral triangle is drawn whose base is removed. If you choose the sides of an equilateral triangle as starting line, in the limit you get the von Koch snowflake curve.  Further “demo movies” can be found here.

This curve was constructed by the swedish mathematician Helge von Koch (1870 - 1924) as an example of a continuous curve in the plane without a tangent at every point. The same curve is also an example of a nowhere differentiable continuous function.

It is important, when talking about fractals, to understand the dynamic processes that create them.  Fundamental to the understanding of a fractal is the process of iteration.  In order to grasp this we must understand the principle of feedback.

Feedback

The process of feedback functions as an iterator.  The same operation is carried out repeatedly, the output of one iteration being the input of for the next one.  Imagine a video camera aimed at the images it produces on a television monitor but twisted slightly so that you get a constantly shifting image.  In order to get a simple intuitive sense of fractal geometry you can extend this process of feedback by imagining a copy machine equipped with an image reduction facility.  The image is simultaneously reduced and multiplied with every iteration.  The standard example is the Sierpinski Gasket.

Self Similarity

One of the defining properties of fractals is built in “self-similarity.”  A figure is self similar when parts of it contain small replicas of the whole.  Every part of a strictly self-similar structure contains an exact replica of the whole.  The most famous example of fractal geometry is the Mandelbrot Set (M-Set).  Benoit Mandelbrot coined the word Fractal.  You can witness some of the extraordinary properties of the M-Set at this site.

Other M-Set sites follow:

The following sites will provide more in depth analysis of fractals as well as fascinating applications in the arts.  All are warmly recommendable.

Earl Hinrich’s Art some very nice things are here

Breugel’s Adoration of the Kings

The Website of JWP 