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A fractal is a geometrical mathematical object which has the property of self similarity. This means that a fractal looks superficially similar on all scales. If you take an image which exhibits this property, and enlarge it, the result looks similar to the original. A true fractal can be magnified an unlimited number of times; each successive magnification having similarities to the previous one. This results in an object which has a non integer dimension (a line is one dimensional, a square two, and a sphere three). The following sequence of images shows one such fractal, namely the Mandelbrot Set. The images were generated with Fractint (see below). The first image shows an un-zoomed Mandelbrot-Set, with a white boxes showing where the following image int he sequence was zoomed from. Notice the bug shape in both the first and last image - superficially, all that has changed is the orientation of the bug. Closer examination shows that there are more miniature tendrils emerging from the bug in Image 5 though. The zoom could be continued much further, but Geocities only give a finite amount of disc space!
Probably the most famous fractal viewed on computers is the Mandelbrot Set, named after Beinot Mandelbrot who first produced images of it in the early 1980s. The zoom sequence above is a Mandelbrot set. Click here to find out how the Mandelbrot set is created.
Other common fractals are Julia Sets - which are similar to Mandelbrot sets, L-systems, IFS (iterated function system). 'New' types can also be created by using a different iterative equation in place of the Mandelbrot Set equation.
Fractals also occur in nature too though, and objects such as ferns, trees, mountains, river courses, clouds are fractal like. A fern is a simple form of fractal, because it can be observed on several scales, and still look much the same. Viewed from a distance it appears like a central branch with several small branches sprouting from it. Closer inspection shows that each small branch has several even smaller branches on it, and so on. A fern only has a limited amount of self similarity - look at it too closely, and there are no smaller branches, hence it is not a true fractal, but a good approximation. (The same limit holds for all physical objects, since the fractal analogy breaks down when the zoom level reaches a molecular scale).
The Mandelbrot iterative equation IS simple, and yet it produces
such intricate images, all from just Z(n+1) =
Zn2 + C.
So why has it only come to light only relatively recently? In fact,
some early fractal work was conducted by Gaston Julia earlier this
century, but it was Mandelbrot who first produced images of these
objects. A closer look at the procedure reveals that it is highly
repetitive (see description of how the Mandelbrot
set is made), with each pixel requiring many iterations in order to
determine if that point lies within the Mandelbrot set or not. It turns
out that even to produce a modestly sized image of 100*100 pixels of an
unzoomed Mandelbrot set requires about 100,000 iterations in total (an
average of 10 iterations per pixel). If we consider that each iteration
requires perhaps 20 computer instructions, then we have roughly 2000,000
calculations for that image. If this were to be done by hand, then
assuming no mistakes, and 5 seconds for each calculation (using an
electronic calculator), then it would take about 7-8 years to complete
the calculations! Without an electronic calculator... Of course, a
modern computer can cough up the data in a fraction of a second, but
modern computers are by definition new inventions, and the sheer mass of
calculations has been impractical to carry out until the electronic
computer revolution of the '60s (the computer on your desk is likely to
be far more powerful than all of the computers in the world put together
in 1960). If you then realise that some highly zoomed, and detailed
images can take a Pentium days or even weeks to compute, you soon begin
to appreciate why iterated fractal calculations are a recent discovery.
There are two ways:
Program your own fractal generators or
Use existing software