"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

Benoît Mandelbrot, 1983
     Euclid versus Mandelbrot      
Fractals are a language, a way to describe a geometry. Euclidean geometry is a description of lines, circles, ellipses and so on. Fractal geometry is described in algorithms - a set of instructions on how to create the fractal. Computers translate the instructions into the magnificent patterns we see as fractal images.

But what are fractals? They are images of the process of a mathematical exploration of the space in which they are plotted.

This image, however, is far more than a flat picture in magnificent colours. A fractal is infinitely complex. That is, if you zoom in on any part of the fractal you will always find more detail. Each stage tends to have the same form as the original. So the fractal lacks scale. A small portion of the fractal is just as detailed as the original.

The amazing thing about fractals is that the formulae used to generate them are often extremely simple. A simple formula can lead to complex images - the Mandelbrot Set is one of the simplest.

Benoit Mandelbrot, in 1979, was playing with a rather simple little equation: (z is mapped as the square of z plus c) where z and c are complex numbers, and c is a constant. The computer screen became the complex plane. Each point on the plane was tested into this equation. The values for the point (like a point on a graph) were fed into the equation:

z z2 + c

This is read: z is mapped from z squared plus c. What it means is:

take the value of z

square it

add c (which is a constant value equal to the position on the screen)
These three steps form the algorithm. This algorithm is then performed over and over and over - that is, it is iterated.

If the iteration went out of control, the point on the screen was plotted white. If it stayed within some bounds, for ever, then the point was considered to be inside the set and plotted black. The Mandelbrot set was born.

Out of this seemingly simple equation and simple steps performed on it arose what some claim to be the most complex mathematical object ever discovered. Many mathematicians are still exploring it, finding more patterns, twirls, filaments, mini-Mandelbrots, dragons and spirals as they plunge deeper and deeper into its depths.

But what about the colours?

Colour was added, indicating how quickly the iteration went out of control. It was found that if the size of z became greater than 2, then it would go on to infinity. So if it took less than 10 iterations to reach 2, the point was plotted as blue, say. If it took between 10 and 20 iterations, then the point was plotted red. And so on. Once it went above 100 iterations and was still less than 2, then it was considered to be inside the Mandelbrot Set. Then the program went onto the next pixel on the screen.

Then, just for fun, programmers made their computers cycle the colours.

So the image needs a computer to exist. It needed the computer to be able to perform the vast number of calculations, to produce the colours at rapid speeds and to allow us to choose any part to explore further.

Mandelbrot coined the word "fractal" to describe his new object and those like it. He argued that the edge of the set was more than a line (of dimension 1) and less than an area (of dimension 2). He claimed it had a dimension somewhere between the two. A fractional dimension - a radical concept!

Fractals are expressed not in primary shapes but in algorithms, sets of mathematical procedures. These algorithms are translated into geometric form with the aid of a computer. The supply of algorithmic elements is inexhaustibly large. Once one has a command of the fractal language, one can describe the shape of a cloud as precisely and simply as an architect might describe a house with blueprints that use the language of traditional geometry.
Jurgens, Peitgen and Saupe, The Language of Fractals.

There are basically three types of fractals, if classified according to the mechanism by which they are generated.

1. Linear Fractals
2. Non-linear Fractals
3. Random Fractals
We will look at a summary of these three major types, and later explore the mathematics in more detail.

1. Linear Fractals

Linear Fractals use only first order terms in the equations used to construct them, that is the variables are not raised to any power above 1.

These fractals are generated using a set of transformations, called affine transformations. This means that any shape which is being transformed is acted on by an equation which will move it to another position, maybe rotate it, or change its size, but straight lines will remain straight.

These fractals are generated by programs which take a point, randomly choose one of a number of transformations available and perform them, then plot the resulting point. This is repeated over and over as the fractal image is built up.

When plotting, these images appear dot by dot over the screen. Sometimes these fractals are referred to as IFS (Iterated Function Systems) due to the system of equations which are used to generate them.

These fractals are also self-similar. In zooming in on the fractal, exactly the same image will be generated. The Sierpinski Gasket, when zoomed, will always look like a Sierpinski Gasket.

2. Non-linear fractals

These fractals include equations with terms which are raised to powers of 2 or more. The most familiar will be the Mandelbrot Set. These fractals are produced by putting the co-ordinates of a point into an equation, iterating according to some algorithm, or set of steps, and plotting the colour of the point according to the resulting behaviour of the iteration.

There are many of them - as many as there are equations to plot! These may not be simple equations like that for the Mandelbrot and Julia Sets but may involve other non-linear forms such as differential equations. Then again, they might introduce their non-linearity using trigonometric functions such as sin and cos. Most of the glorious fractals you will see in a web search for fractal images are non-linear.

3. Random Fractals

These fractals are plotted in a completely different way. Points are allocated values randomly, such as colour or height, but then tempered by the values of other points near to them. For example, the point between two existing points may be raised by a random amount from the average of the two points nearby. The colour of the point may be chosen randomly and then averaged with values based on the nearby points and their distance away.

These fractal images can look very like clouds, if colours are used, and mountains, if heights are used. Or the colours of a cloud can be transformed into heights, and - presto - a mountain.

If the fractal is random, a determining formula can't be given. It will have to be described. But more of that in the task Random Fractals - Plasma.

Go Search! Use an image search in your favourite search engine and hunt for fractals. You will find millions of stunning images on the Internet.
Most will be non-linear fractals. See if you can identify any linear or random fractals.
You may want to refine your search with any of the key words raised in this task. Try it! See what you find.

In the Good-Old-Days we had these really slow computers. OK, things took much longer, but there was one really big advantage - you could watch the computer plot the points at a speed you could cope with.