|
|
|
 |
Home page | |
 |
|
New pages will be added soon
If you wish to receive an e-mail when these pages will be updated,
look at here.
| JPG format with low compression rate is used in order to keep a high quality rendering.
Each of the next pages is made of 12 reduced pictures that will enable you to display the original pictures in 800x600 (with a few exceptions). Owing to the large size of these files you can also display 640x480, more compressed images. These sizes are acceptable compromises between quality and time needed to download the images. The fractals made with Fractint have recently be redrawn at 2048x1536 size (with the few exception of 3 fractals because I have some problems with the formulae), then resized to 800x600 and 640x480 using anti-aliasing in order to improve the rendering of the images. For technical reasons only the latest images made with Fractal Orbits have been redrawn, but it was not possible to draw initial images greater than 1024x768. |
The
pages of the album
Are these pictures fractals ? |
|
| This question is not as paradoxical as it seems at first glimpse. To take the two classical
examples of the Julia and Mandelbrot sets, a mathematician only has to determine if the point
belongs to the set or not, and to give it a colour (black in the examples in the previous pages) The other points are outside and, at first approximation, are not interesting to him. Nevertheless they convey an extra piece of information : after how many iterations does the function diverge for the coordinates of this point ? It is possible to attribute a colour to each dot of the picture, which is function of the number of necessary iterations to observe the divergence. Rigorously speaking a grey colour range is enough. Nevertheless the picture is more attractive, even if it doesn't add any information, if we use a colour map. Of course the mathematician will try to compute with a big enough number of iterations for the picture to use best the resolution of the screen. With a number of iterations that is insufficient, details can be lost ; with too many, it will take the computer much longer but the precision will be illusory because of the limited display characteristics of the screen. Nevertheless, even when one takes these precautions, the picture can be deceptive. In the previous pages it is easy to notice small islets that seem isolated from the main body of Mandelbrot's set. And yet mathematicians tell us that this set is connected. In fact the "threads" that link the islets to the rest of the set are too thin to show on the picture and one has to fake these picture, by magnifying artificially these details so that they can be seen. Why are we entitled to wonder about the fractal nature of some pictures shown ?
|
|
The colour palettes |
|
|
A first point is the choice of colour maps. Since it is possible to spread out these colours into
successive stripes, in the image will appear some structures that do not exist as such.
By sophisticating this some more, it is possible to make the colours of the various stripes increase
or decrease regularly (in a linear, sinusoidal, logarithmic... way) on each side of a saturation
value or of maximum (or minimum) brightness. Under some conditions a certain pseudo-relief could
appear, but with no connection with reality.
|
|
"Fiddling" the calculations |
|
| Another source of modification relies in the mathematical treatment of the used function. One
can, for example, limit the number of iterations so as not to visualize all the
details of the picture and make it simpler. Here again we have a heretic mathematical manipulation. It is also possible to modify the test that enables the computer to detect the divergence of the function. By taking again the familiar examples of Julia and Mandelbrot sets, the classical test consists in checking whether |z|<=4 (for Fractint |z| is the squared modulus), but one can sometimes obtain spectacular results with lower values. This leads to the exclusion from the set of points that belong to it, another mathematical heresy. Many other complex mathematical treatments of the image are possible.
|
|
As a conclusion |
|
| In a certain number of cases the self-similarity that characterizes fractals is very limited, or even absent from the final picture because of the methods mentioned here above. This is why I personally prefer talking of graphical art based on fractals, rather than of fractal pictures. However, the only possible modifications are algorithmic treatments applied to the whole image (various anamorphosis or rotations, for instance), work on the colour palette or anti-aliasing. In some of the following pages I show the various possible results obtained from the same basic image. None of the pictures presented have been manually modified in their pixels. | |
|
To leaf through the fractal album |  |
|
|
Home page | ||
 |
Updated : 07/10/02 |
||
