Fractals

A member of the mandelbrot set

Introduction

Fractals are beautiful diagrams, formed from some more complex chaotic maths. An inherent feature of fractals is self repitition. Self repetition is where a small part of the structure, enlarged, looks like the whole structure. There are many different types of fractal, and each type yields an infinite number of fractal diagrams. This is because fractals are infinitely complex, with a suitable program you can zoom in to any magnification, and the structure will remain as complex as before the zoom. Fractals utilise complex numbers. A complex number has two parts, one real, one imaginary. The real part is a normal number, whilst the imaginary part is a multiple of the square root of -1, represented by i. The complex plane is the plane with x-axis comprising the real part, and the y-axis comprising the imaginary part. In this way, any complex number can be plotted on the complex plane. All the fractal pictures below were produced with Fractint, a free fractal producing program.

The Mandelbrot Set

This set was devised by "the father of fractals", Benoit Mandelbrot, and is the original fractal set. The fractal is plotted on the complex plane. Then, to plot the fractal, we take each point on the complex plane in turn. This is called c. Then, starting with z₀ = 0, we use the iterative formula:

z_n+1 = z_n² + c

According to how fast this iteration leads to z > 2, the point c on the complex plane is shaded a certain colour.

The Julia Set

This set is named after Gaston Julia, a French mathematician. The Julia set is plotted on the same axis as the Mandelbrot set, and is closely linked. It has a very similar formula. For each c, z₀ = c, and then the same iterative formula is used:

z_n+1 = z_n² + c

Again, according to the behaviour of this iteration, the point c is coloured.

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Will Bolam 2001