IFS Fractals

Preceding Page
fougère

Since IFS fractals have only a little part in the discussion about the relationship between Fractals and Art, this page may be considered a parenthesis, a kind of footnote that the hurried reader may skip, but it also (hopefully) contains informations of interest for anyone curious of the variety of the fractal world. There will be no mathematics other than intuitive notions of geometry.

An IFS fractal, in the sense of Barnsley, is defined by a set of elementary geometric transformations which are called "linear" or "affine" by mathematicians. In the everyday language, they are a combination of

  • translations
  • rotations
  • linear compressions along the vertical or horizontal axis, or both. The compression ratios along the two axes can be different.
  • vertical or horizontal shears, such that a rectangle is transformed into a parallelogram through the sliding of a side along itself.
The only requirement is that the transformations must be contractive, i.e. the distance between two points must decrease (at least, not increase) in the transformation.

The transformations to be implemented in the IFS set depend upon the figure to be redrawn. There is a magical precept that must be satisfied: if the target figure is transformed through the various transformations in the set, one must get exact parts of this figure, and superimposing all these parts must reconstruct the whole figure. The preceding fern thus asks for 4 transformations, as illustrated below.

IFS
The result of each transformation is shown in blue. The last transformation is a complete compression of the fern along the piece of stem which was not covered by the three first transformations. Clearly, this splitting of the fern is based upon its strong self-similarity, which makes every leaf an exact reduction of the whole fern.

There is something magical in redrawing the figure. First there is an initialisation stage, where an arbitrary point (for instance the center of the screen) is moved through the various transformations in the set, in a large random sequence (say 1000 transformations or more), with nothing written on the screen. Then the random sequence is continued, but now with writing every transformed point. And the miracle happens, the fern appears, pixel by pixel!

One can understand a bit what happens by noticing that each of the transformations has invariant points (i.e. points which do not move in the transformation) and that these points necessarily belong to the figure. Indeed, in the random sequence, the same contractive transformation may occur several times in succession, and, since it is contractive, one goes closer and closer to these invariant points. Therefore, sooner or later, the moving point enters the figure and this is the aim of the initialisation stage. Then, according to the chosen transformation, the point wanders here and there in the figure, more or less uniformly depending on the probabilities given to the various transformations in the random sequence.

 

fern2

 

  Several IFS initiation programs have been proposed after 1988, including mine for the Amiga, after M.F. Barnsley et A.D. Sloan had published their popular paper "A Better Way to Compress Images" in Byte, (January 1988). I was rapidly convinced that these tools were marvelous for making ferns, even rather strange species –as shown on the left,– but of little interest for anything else, specially anything artistic, at least with what was available in the public domain. And that IFS "trees" were rather unrealistic...

And what about colour? In these elementary programs, the only recipe was to change the color every time that a pixel was written again along the random sequence (Barnsley himself suggests nothing else in his book Fractals Everywhere). The preceding picture was obtained in this way. To go a bit further, one could just superimpose several quasi-monochrome pictures, as shown below.

fern1

fern3

 

 

However, when I look at this last picture, obtained in 1989 by mixing the details of two "ferns", I still feel some regret for not having explored further the textures which can be obtained in this way. But the tool was really lacking flexibility and the machines of the time, even the Amiga, were still too primitive.

Preceding Page


[ Return to home page ]    [ Art and Fractals Contents ]

Discovering the fractal world:
Introduction - Mandelbrot Exploration - Lyapounov Exploration - Von Koch Curves - IFS Fractals - Fractal Dimension - Mandelbrot Relatives - Finest Fractal Pictures - Software - Biblio and Links

Fractals and mysticism:
Introduction - The Mysticism of Infinite - Non-Euclidean Art?

A new Art?
Introduction - Fascination of Fractals - Fractals and Photography - Definitions of Art - The Colour Choice - Other Colour Choices - Fractalists Painters - Compositions with Mandelbrot - Put a pretty girl - Algorithmic Art - Beyond the "Fractal" Art