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Fractal Information

   

 

CONTENTS OF THIS PAGE

What are fractals?
How were fractals named?

Why are fractals appealing?
Generating fractals yourself


What are fractals?

A fractal is a geometric construction that exhibits self similarity across all scales. In mathematical terms it does not matter how far you zoom into a fractal image there will be an equivalent level of detail at every stage. Fractals are also found in the natural world. Many phenomenon which are difficult to analyse using traditional methods can be modelled using fractals eg clouds, trees, landscapes, coastlines etc.

Although mathematical fractals contain an infinite level of detail with 'real world' fractals this is not the case, for example you can magnify the view of a coastline and at first there will be major estuaries and larger bays. Further magnification will reveal smaller estuaries and bays then down to coves and inlets and finanlly individual rocks and pebbles. At some point further magnification causes the previously observed fractal structure to breakdown. Fractals in the natural world are subject to this sort of limit. However because they have fractal structure over a range of scales fractal techniques can be used to successfully model them.

Limitations also effect the computer generation of fractals. Even though the mathematical processes used as the basis of calculation are capable of being magnified forever without a loss of detail, a calculator, computer or (heaven forbid!) pencil and paper are not. When an image is calculated on screen it is limited to the resolution of the monitor therefore this process only provides a representation of the fractal. You could magnify and recalculate so that a small region of the previous image is now displayed on the monitor as the new image. At least this will show levels of detail which were not previously apparent but nevertheless any particular render will only be a representation of the fractal at a particular scale.

Another feature of fractals related to self-similarity is 'fractal dimension'. Going back to coastlines, how long is the coastline of say, Britain? Unfortunately, there is no straightforward answer as it depends on the size of the ruler used to measure it. The smaller the ruler the more details can be measured and the larger the answer will be. Measurements for different ruler sizes can be used to calculate the 'fractal dimension'. A coastline will have a fractal dimension between 1 and 2, compared to the topological dimension of 1 that lines have. All fractals will have a fractal dimension greater than the topological dimension.

How were fractals named?

The term 'fractal' was coined by the Benoit B. Mandelbrot in 1975 to collectively describe sets that although previously studied by other mathematicians had, at that time, remained unnamed. Effectively this was the birth of Fractal Geometry. Mandelbrot derived the term 'fractal' from the Latin for 'irregular'. Mandelbrot later asserted that fractals deserve to be called 'geometrically chaotic'.

Why are fractals appealing?

There is no entirely straightforward answer. Fractal geometry is very much at odds with the Euclidean geometry which we see around us in our man made environment; straight lines, clean curves, smooth surfaces, perfect parallels and perpendiculars. These are not the stuff of nature, they have no aesthetic 'depth'. No one wants to see a park or garden consisting of cones and triangles, it might be argued that this is because we enjoy plants and trees as an 'break' from the simple, tidy geometry of everyday life, but does it really go further than this? A smooth polished surface has a certain attraction but the attraction is much greater if it is smooth polished marble or beautifully grained wood. Not that it is suggested that complete disorder has any appeal, untidiness appeals to no one. As with marble or wood grain the beauty of fractals lies in their 'order within disorder'. For my own part the most attractive fractal images are those where the longer you look the more you see, what starts off just looking like an irregular shape will have form which will be repeated at smaller scales, not necessarily identical but similar. But, as in all matters of artistic taste, individual opinions will vary considerably.

Generating fractals yourself

A further appreciation of fractals comes from generating them yourself. It is possible writing your own programs but a much easier way is using either a freeware or shareware program. In many cases using a such a program means you can avoid all the difficult maths stuff if you wish and just concentrate on producing great images. But remember, as with all things that are worthwhile, it does take a bit of practice and perseverence to produce good results. Further details of such programs can be found on the Links page.