Jean-François COLONNA
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CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
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(CMAP28 WWW site: this page was created on 06/07/1995 and last updated on 11/14/2023 17:56:46 -CET-)

(published in The Visual Computer, Volume 11, Number 8, 11/1995)

Abstract: The human intelligence and in particular its creative potential seems to be, for many scientists, made of processes that cannot be automated by the means of a computer. This WWW page tries to show, as already exhibited with the fractal geometry, that the iteration and the combination of very elementary operations can give very complex behaviours and shapes. It will give some practical examples in order to encourage reader involvement and new experiments in particular during classroom settings.

Keywords: accumulation, brain, chaos, conformal mapping, convolution, creativity, Fast Fourier Transform, fractal geometry.


Whether the human brain is a machine or not is one of the oldest questions posed by human beings. Two answers are given: yes and no, neither being scientifically justified. My personal view about this question is that the brain is a machine that constantly receives external data (by the means, for example, of the visual system), transforms, processes and memorizes it, and then produces output at the muscular level (to walk, to talk,...). Creativity could then be the "simple" result of numerous combinations and transformations of former memorized data thus giving birth, from time to time, to very surprising new data...

I should like to illustrate this personal point of view by means of an analogy using very simple computer graphics algorithms.

Genetic algorithms [01], the fractal geometry and the deterministic chaos [02][03][04] have already exhibited the fact that iteration and combination of operations can result in very complex behaviours and shapes. The main goal of this WWW page is to show this remains true even with the most elementary operations. Moreover, it should be noticed that most chaotic processes are sensitive to rounding-off errors (see [05] and one of the author's WWW pages); the consequence is that their results may depend on many extra-parameters:
Then, the user could be faced with impossibilities: for example, to reproduce results obtained before upgrading his or her computer.

As suggested above, let us describe a few simple tools for false color pictures (the extension to true color pictures is obvious); figure "Four tool pictures" displays four very simple "seed" pictures useful for our demonstration.

Once again, my purpose here is not to exhibit very new and complex algorithms, but only to remind the reader that simple processes, when iterated and combined together, can give birth to very complex and unpredictable shapes (in the computer graphics realm) and behaviours. The figure "Synthesis of tridimensional stained-glass window" displays four frames from an animation produced using almost all the preceding processes. It can be described as:
                    G(0)   = GRID
                    P(t)   = B(GF(GP(GF(SCM1(G(t)),GAUSS),GF(SCM2(G(t)),GAUSS),STAR),GAUSS),WHITE)
                    G(t+1) = ELG(G(t))

with t [0,127]
(when in fact it is a little more complicated for it is a true color one). The shapes then obtained surprised me, even though the computer did nothing more than to execute my requests; as stated earlier, the combinations and the iterations of simple processes (using simple "seed" pictures) give birth to very surprising pictures (I dare not say beautiful), some of which are even "biological". This must be a lesson for us: simplicity can produce complexity, and then surprises for the observer. My surmise is that when an artist creates such pictures ("by hand"), they are only the result of numerous combinations and transformations of preexisting elements stored in his memory (each coming either from his five senses or from former "computations") and nothing more.

The algorithms described in this WWW page are elementary and then easy to implement by novice programmers, when the starting pictures "Four tool pictures" (WHITE, GAUSS, STAR and GRID) suggested here are very simple. I should like to encourage reader involvement, in particular in classroom settings, with experiments (about starting pictures, kernel pictures, parameters,...) giving birth to the huge variety of shapes and textures needed for our future virtual worlds...

Copyright © Jean-François COLONNA, 1995-2023.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 1995-2023.