Guided Tours of Buried Galleries
(inside a Computer)






Jean-François COLONNA
www.lactamme.polytechnique.fr
jean-francois.colonna@polytechnique.edu
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France
france telecom, France Telecom R&D

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(CMAP28 WWW site: this page was created on 11/13/2001 and last updated on 01/28/2014 15:11:02 -CET-)



[en français/in french]


(Arts et Mathématiques, International Conference, Maubeuge, 09/20-22/2000)



Keywords: Anaglyphs, Art and Science, Artistic Creation, Autostereograms, Celestial Mechanics, Computer Graphics, Deterministic Chaos, Fractal Geometry, Intertwinings, Mathematics, Natural Phenomenon Synthesis, Numerical Simulation, Physics, Quantum Mechanics, Rounding-off Errors, Scientific Visualization, Sensitivity to Rounding-Off Errors, Software Engineering, Stereograms, Texture Synthesis, Virtual Experimentation, Virtual Space-Time Travel.




1-INTRODUCTION:

The title of this text can seem quite mysterious, but it reveals the double reading it is possible to make of the pictures we will describe. As a matter of fact, most of them are computed in order to be used as scientific or pedagogic tools ; but that does not exclude to carry them out under aesthetic constraints while being attentive with proportions, with colors,... even from time to time by forgetting Science. The word gallery must thus be understood with to quite different meanings which will be defined hereafter.




2-ART GALLERIES:

The art galleries buried inside computers are those which offer pictures for the only pleasure of the eyes, by avoiding the description of the underlying mathematical proceses. But which is then the status of these pictures: are they works of art in the classical sense of this term? It is advisable to recall initially that they lose here a fundamental property (that gives part of their values): the one of uniqueness; with the pictures known as digital, any copy is as perfect as the original. They lack also a noble medium: here nothing like the Carrara marble... But beyond these negative characteristics, there is a point which seems very promising: the ermergence of the concept of potential work of art: the work of art is the tool. As a matter of fact, I am myself an "algorithm conceptor" and the result of the creative process is not one or more pictures produced by a certain program, but rather well this last one (the program) which thus must be seen as the potential container of a quasi-infinite (since the infinity does not exist for a computer) number of pictures of the same type. For instance, this is the case of a N-dimensional fractal field generator I have conceived; it is able to produce an incredible variety of natural phenomena: mountains, clouds,... This concept, which Jorge Luis Borges would certainly have loved, is unfortunately difficult to explain to the general public and thus it is mandatory to produce some "objects" more easily communicable (some pictures) that will be exhibited in actual Art Galleries and more often (unfortunately?) in virtual ones.




3-MINE GALLERIES:

Beside the Virtual Art Galleries buried inside computers, more mysterious Mine Galleries furrow the memories of our digital machines. A mine, in its most common sense, is a place where one digs the ground in the hope to discover wealthes or even treasures; but what is this furrowed ground here? It is made of Mathematics which is a fundamental element structuring our perception of the Reality; these Mathematics have become during the past centuries the language of paramount importance with which Science describes the studied phenomena. Even if this language is certainly not the Ultimate Reality, but rather well a reflection of our own cognitive processes, it enabled us to imagine the Infinity and to see beyond the horizon...


Virtual Experimentation is a recent scientific approach; it starts with the mathematical model of a certain physical phenomenon, then, in order to solve the equations it contains, some so called numerical methods must be applied. At last, all this is translated into computer programs that, after debugging, are exploited in order to produce results. Most of the time, their analysis is not obvious due to the large quantity of digits they contain; this problem can only be solved using the techniques of picture synthesis. Thus, modifying (and playing) with the model parameters and seeing the model evolution on a screen is a new kind of experimentation.


This approach is very promising as well at the fundamental scientific level as at the industrial one. One must remember the prophetic words of Heinrich Hertz that said during the nineteenth century: "one cannot escape the feeling that these mathematical formulas have their own existence, that they are more erudite than those who discover them and that we can extract more science from them than it was input at their creation". This is also true for the virtual experimentation, as well for the models as for the programs used, but on the very important condition of knowing well the dangers and the limits it contains.


With regard to the mathematical and computational aspects, it is obvious that the real numbers are fundamental. By the way, this omnipresence is quite mysterious: as a matter of fact, for instance when speaking only about distances, what is the usefulness of the infinite precision of the real numbers, whereas the Universe seems to measure only 1.5e61 (using the Planck length as the measurement unit)? Their use seems to come to the need to have differential equations that are obtained by the means of the manipulation of infinitely small numbers. But (unfortunately) a computer is only a finite machine (its memory, although more and more important, is physically limited) and the manipulated values must be sampled. Thus, for these machines, the infinity and the continuity do not exist, whereas they are essential to us! One of the consequences of this fact is that some fundamental properties are lost: this is the case for the associativity of the addition and of the multiplication of numbers. To forget that can lead, in certain circumstances (case of the problems known as sensitive to initial conditions), to some nuisances (euphemism...).


The picture synthesis, contrary to a naive intuition, is also source of difficulties. As a matter of fact, we are here well far from the applications praising the qualities of a car or of a package of detergent. The so called Scientific Visualization has very often to exhibit abstract objects (mathematical structures for instance) and thus without a priori images. But does that want to say that each "natural" object has its own image? Unfortunately not: this is obvious in the quantum mechanics realm, but more close to us, what is the color of a pressure field? This question could seem absurd, but it must be answered in order to display that kind of results ! The morality of all that is quite simple: the visualization of scientific "objects" is in most cases arbitrary and thus subjective. Then it very simple to build "orthogonal" (inconsistent) views for a given object.




4-FOLLOW THE GUIDE:

The techniques of the Virtual Experimentation when using relevant (and refutable) models, robust numerical methods, reliable programming (maybe the most difficult task...), fast computers and at last the most neutral picture synthesis, make possible the Virtual Space-Time Travel. The exploration of all known scales, from the elementary particles to the whole universe, become possible:


5-CONCLUSION:

The Mathematics, benefitting from the fabulous progress of the Computer Science, both play of the spirit and an open window on Reality, are also simultaneously a fabulous object and a tool for creation: object because they propose to the artist new sources of inspiration ; but also tool because they give to him, just like to the scientist, new means of expression that can be masterized on condition their limits are well known.


Copyright (c) Jean-François Colonna, 2001-2014.
Copyright (c) France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2001-2014.