What Are Fractals?

A fractal is defined as a geometric shape that is complex and detailed in structure at any level of magnification. 

Often fractals are self-similar—that is, they have the property that each small portion of the fractal can be viewed as a reduced-scale replica of the whole. 

 One example of a fractal is the "snowflake" curve constructed by taking an equilateral triangle and repeatedly erecting smaller equilateral triangles on the middle third of the progressively smaller sides.

Theoretically, the result would be a figure of finite area but with a perimeter of infinite length, consisting of an infinite number of vertices. 

A fractal cannot be treated as existing strictly in one, two, or any other whole-number dimensions. 

Instead, it must be handled mathematically as though it has some fractional dimension. The "snowflake" curve of fractals has a dimension of 1.2618.

Recursion in the Complex Plane

As seen from the Koch ("Snowflake"), recursion can create some beautiful mathematic images.

The most famous of these is the Mandelbrot set, created by iteration z(new)=z(old)^2+c, where z and c are complex numbers. 

A complex number is one in the form a+bi, where i is denoted as the square root of -1. Here is a series of images of the Mandelbrot set:

The mandelbrot set is a fractal, exhibiting many interesting properites.

You can run many different types of recursion to get neat sets. All of these sets are called Julia Sets.

Visit James Hestridge's site for a Mandelbrot Set Applet. The source code can be obtained here.


http://www.mtholyoke.edu/~cagora/proj2/whatare.htm