What is a fractal?

A fractal is simply a self-similar image, ie if you enlarge a bit of it then you get something similar to the original image.

It is a computer-generated image that graphically represents the behaviour of a mathematical equation. It is like a snapshot of a mathematical object that would otherwise be invisible.

Paul Lee writes, "Fractals are extensions of traditional Euclidean shapes, such as lines, squares, and circles, with two fundamental properties. First, when you view fractals, you can magnify them an infinite number of times, and they contain structure at every magnification level. Second, you can generate fractals using finite and typically small sets of instructions and data."

Here is the definition given in a FAQ: "A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale."

There is a high-quality explanation of fractals on wikipedia.

Here is my attempt to give a precise explanation of fractals with the absolute minimum of maths.

What is special about fractals?

The two features that stand out for me are their complexity and "self-similarity". The latter refers to the fact that when a fractal is enlarged ("zoomed into") then the resulting image is usually similar to the previous one, sometimes almost identical.

Note that a fractal has infinite granularity in that one can zoom in without limit (at least in theory) to show ever finer detail. This is equivalent to making the spatial range smaller and smaller.

Fractals mimic some of the shapes found in nature. As in nature, fractals rarely have any straight lines. The most typical shapes are the spiral and the Mandelbrot - a circle with a dip, with smaller circles hanging off it.

The other, and for me the most important, feature of fractals is that some of them are beautiful.

Why make fractals?

Two motives for making them are the search for beauty and simple curiosity. Making fractals is a way of exploring the natural world, akin to collecting crystals or sea-shells, only far richer and more active. You may be the first person in the world to see a new and fascinating shape. It is as though a non-human artist is drawing images for you, unconstrained by any human notions such as subject/background, balance, perspective or colour combinations.

For me, the appeal is to create an image with new and interesting structure. I see fractal-making as entering new worlds unknown to man (or even woman).

Who invented them, when, where and why?

During World War I, a French mathematician called Gaston Julia was probably the first to draw a fractal. Benoit Mandelbrot is normally seen as the father of fractals. He played a pivotal role in the development of the branch of mathematics called "chaos theory", the study of very complex dynamical systems. Such systems are characterised by sensitive dependence on small changes,ie a small change can cause a surprisingly large effect. Fractals are used to model the behaviour of complex and almost unpredictable systems such as the weather and earthquakes.

Where are fractals seen in nature?

Fractal shapes are seen in fern leaves, sea-shells, snow-flakes and coast-lines.

How are fractals made?

They are made using computer programs such as Tierazon, Sterling, Fractint or Ultrafractal.

How long does it take?

About 1 to 5 minutes for a full-screen fractal, but this can vary widely, depending on screen resolution, the program used, the equation, and other factors.

How do I start?

First choose a fractal program that you want to use - click here if you want to read a description of a few of the many fractal-generating programs available on the Net, mostly for free. Then download the program, install it on your machine and run it.

Just play with the controls of the program until you get an image that you like. You can then save the image on your screen to a disk file that can be stored permanently and shown to others.

How is a fractal calculated?

Here is a very simplified explanation of how a fractal is actually generated.

Each point on the screen is assigned a pair of values, such as ( 1, 2 ), giving its horizontal and vertical position. This pair of numbers is fed into an equation and the resulting pair of numbers, such as ( 7.235, 4.001 ) is fed back into the same equation. This process is repeated many times.

How does this create an image? you ask. The answer is that the size of the final value, after say 200 iterations, is used to select a colour for the dot at that point on the screen. This process is repeated for every point on the screen. No wonder it takes a while.

If that was too complicated, try this explanation by Dan Turner: "Fractals are cool pictures generated by a freaky mathematical function crunched by a computer program."

How is a fractal really made?

If you really want to see how a fractal-generating program works then download the source, which includes a detailed explanation in clear English, and executable of Winmand. This is about as simple as a Windows fractal generator can be.

Here is the detailed explanation.