Jean-François COLONNA

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

[

[

[

(CMAP28 WWW site: this page was created on 08/06/2015 and last updated on 09/12/2020 14:33:02 -CEST-)

To make progress in our knowledge of the Universe, we must do Mathematics: juggling with formulas, solving equations,... All this is generally abstract, but nothing forbids us to use the sense that evolution has given us and particularly the vision; our eyes are made to be surprised, and today the computer allows us a new experimental approach: the virtuality. A so-called

THen what to do with all these numbers? Obviously reading them would be a non sense. Is there an objective way to exhibit them? Their display as pictures seems to be the best solution: as a matter of fact our visual system is a high bandwidth processor that can instantaneously react to surprises. This translation of numbers into pictures implies the existence of an universal code. But is it true?

On the one hand almost all our display systems (sheet of paper, screens,...) are bidimensional. When displaying sets of dimension 3 and higher, projections will be mandatory thus hidden their full complexity. But we are accustomed with this process for it is the case with protography. No problem with familiar objects: everybody has seen mountains (fractal ones...) , but with unfamiliar objects like this unkonwn structure its understanding is more difficult. Beyond the third dimension, the situation will worsen and rotations or again cross-sections will be used as with this Julia set computed in a 8 dimensions space.

On the other hand many measures (real or virtual ones) do not have pictures at all (a pressure field for example). Moreover, pictures could be sometimes forbidden: this is the case with the Quantum Mechanics that does not allow this picture of a nucleon since positions, velocities, shapes or again colors have no meaning there.

Nevertheless one could believe that with very simple sets, objective representations could be devised. Let's recall that a numerical screen is a bidimensional array of numerical values (colors and luminances). Then the display of a matrix should be objective. Unfortunately, this example shows it is false. As a matter of fact to visualize this array one have to answer to stupid questions like: "what is the color of 23?". Obviously the answer is arbitrary: in this picture four different sets of colors are used and the four sub-pictures then obtained seem to exhibit four different objects and worse, four incompatible objects (see the discontinuity and the periodicity that appears and disapppears at will...). Then one can hide that exists and one can exhibit that does not!

Thus if the objective display of a bidimensional array is impossible, it is obvious that the objective display of more complex sets is impossible...

At last it is useful to remember that these pictures are computed to be viewed by people. Then one must not forget that our visual system is subject to optical illusions. For example the one of

Then objectivity and unicity do not exist for the display of numerical sets. Nevertheless, it is useful to devise some useful practical rules: to prefer simple pictures rather that complex ones, to use the so called

Beside these difficulties one must not forget the ones inherent to the use of computers: programming is very difficult or again the fact that numerical computations are not safe. Nevertheless scientists and engineers have in their hands new tools that will be as revolutionary as the microscope and the telescope in their times and a wonderful opportunity for an encounter between Art and Science .

Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2015-2020.