#
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
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**[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.