GALLERY:

NON DETERMINISTIC FRACTAL GEOMETRY
AND
NATURAL PHENOMENON SYNTHESIS

[Géométrie Fractale Non Déterministe
et
Synthèse de Phénomènes Naturels ]

Jean-François COLONNA

CMAP (Centre de Mathématiques APpliquées), Ecole Polytechnique, CNRS
france telecom, France Telecom R&D
91128 Palaiseau Cedex
France

tel = +33.(0)1.69.33.46.45
jean-francois.colonna@polytechnique.edu

http://www.lactamme.polytechnique.fr

http://www.cmap.polytechnique.fr/~colonna

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(this page was created on 03/15/2000)
(this page -belonging to the CMAP28 site- was last updated on 03/01/2010 20:02:12 -CET-)

• Principles [Principes]:
• Clouds [Nuages]:
• Landscapes [Paysages]:
• One Landscape and Various Meteorological Conditions [un seul Paysage dans des Conditions Météorologiques Variées]:
• How the West was won [la Conquête de l'Ouest]:
• Fractal Euclidian Landscapes [Paysages Fractals et Euclidiens]:
• Landscape Interpolation [Interpolation de Paysages]:
• Deep Space [Au plus Profond de l'Espace]:
• Some of my first fractal pictures made in 1980 -very low resolution: 256x256, 8 colors- [Quelques une de mes premières images fractales faites en 1980 -très basse résolution: 256x256, 8 couleurs-]:
• Tridimensional Structures [Structures Tridimensionnelles]:
• Bidimensional Erosion Process according to the Sapoval-Baldassarri-Gabrielli Theory [Processus d'Erosion Bidimensionnel suivant la Théorie de Sapoval-Baldassarri-Gabrielli]:
• Bidimensional Etching Process according to the Sapoval-Kolwankar Theory [Processus d'Attaque Chimique Bidimensionnel suivant la Théorie de Sapoval-Kolwankar]:
• Tridimensional Erosion Process according to the Sapoval-Baldassarri-Colonna-Gabrielli Theory [Processus d'Erosion Tridimensionnel suivant la Théorie de Sapoval-Baldassarri-Colonna-Gabrielli]:
• Diffusion Processes [Processus de Diffusion]:
• Bidimensional Brownian Motion [Mouvement Brownien Bidimensionnel]:
• Tridimensional Brownian Motion [Mouvement Brownien Tridimensionnel]:
• Bidimensional Diffusion Limited Aggregations and Fractal Aggregates [DLAs et Agrégats Fractals Bidimensionnels]:
• Fractal Diffusion Front in a Bidimensional Medium [Front Fractal de Diffusion dans un Milieu Bidimensionnel]:
• Fractal Diffusion Front in a Tridimensional Medium [Front Fractal de Diffusion dans un Milieu Tridimensionnel]:
• Tridimensional Diffusion Limited Aggregations and Fractal Aggregates [DLAs et Agrégats Fractals Tridimensionnels]:
• Iterated Function Systems (IFS) [Systèmes de Fonctions Itérées (IFS)]:
• Recursive "Living" Structures [Structures "Vivantes" Récursives]:

Non Deterministic Fractal Geometry and Natural Phenomenon Synthesis [Géométrie Fractale Non Déterministe et Synthèse de Phénomènes Naturels]:

Principles [Principes]:

• The iterative process used to generate bidimensional fractal fields (16 iterations).
• The iterative process used to generate fractal mountains (16 iterations and bird's-eye view).
• The iterative process used to generate fractal mountains (16 iterations).
• Bidimensional clouds or height field.



Clouds [Nuages]:

• Light cloud dynamics -this sequence being periodical-.
• Light cloud dynamics -this sequence being periodical-.
• Light clouds.
• Light cloud dynamics at sunset -this sequence being periodical-.
• Light clouds at sunset.



• Zoom in on light clouds.
• Zoom in on self-similar light clouds.



Landscapes [Paysages]:

• Zoom in on mountains (with 3D clipping, bird's-eye view).
• Zoom in on self-similar mountains (with 3D clipping, bird's-eye view).



• Mountains at sunrise.
• Animation of a sunrise on mountains.
• Mountains at sunrise.



• Fractal synthesis of mountains with vegetation and clouds.
• Fractal synthesis of mountains with vegetation and stormy clouds.
• Snowy multifractal mountains.
• Snowy multifractal mountains.
• Snowy multifractal mountains.
• Snowy multifractal mountains.



• Mystery mountain.
• Mystery mountain at sunrise.
• Bloody mountain.
• Pinky mystery mountain.
• Meteor Crater.



• Archipelago.
• Foggy Islands.
• Islands at sunset.



• Cloud dynamics -this sequence being periodical, bird's-eye view-.
• Mountains and light clouds (bird's-eye view).
• Mountains and light cloud dynamics -this sequence being periodical-.
• Mountains and light clouds.



• A set of 4x3 stereograms of a fractal mountain.



• Earthquake (bird's-eye view).
• Earthquake.
• Earthquake.
• Earthquake with multifractal mountains.
• Earthquake with multifractal mountains.
• Periodical earthquake with multifractal mountains (bird's-eye view).
• Periodical earthquake with multifractal mountains.
• Periodical earthquake with multifractal mountains (bird's-eye view).
• Periodical earthquake with multifractal mountains.



• Dynamics of an erosion process (bird's-eye view).
• Erosion process.
• Dynamics of an erosion process with fog.



• Long term evolution of a landscape (bird's-eye view).
• Long term evolution of a landscape.
• Synthesis of periodical tridimensional textures by means of a fractal process.



• Tridimensional texture animation by means of the Generalized Product (bird's-eye view).
• Synthesis of tridimensional textures by means of a fractal process.
• Synthesis of tridimensional textures by means of a fractal process.
• Synthesis of tridimensional textures by means of a fractal process and of a fourth degree Verhulst-like dynamics.



• Mountains with slight linear structures and discontinuities (bird's-eye view).
• Mountains with slight linear structures (bird's-eye view).
• Mountains with slight linear structures (bird's-eye view).
• Mountains with large linear structures and discontinuities (bird's-eye view).



• Multifractal mountains (bird's-eye view).



• Waltzing snowy Rocky Mountains.
• Snowy Rocky Mountains.
• Snowy Rocky Mountains.



• Grand Canyon (bird's-eye view).
• Foggy Grand Canyon.
• Opencast mine (bird's-eye view).
• Foggy opencast mine.



• The erection of the Babel Tower.
• The erosion of the Babel Tower.
• A set of 4x3 stereograms of a few Babel Towers.
• Foggy Babel Tower.



• Foggy Cliffs.
• Mystery cliffs at sunrise.
• Foggy Cliffs.
• Foggy Cliffs.



• The tide.



One Landscape and Various Meteorological Conditions [un seul Paysage dans des Conditions Météorologiques Variées]:

• Painted Desert.
• Mountains and clouds at sunset.
• Mountains and clouds.
• Mountains and cloud dynamics -this sequence being periodical-.
• Mountains and clouds.
• Mountains and fog.
• Mountains and fog.
• Mountains and snowfall.



How the West was won [la Conquête de l'Ouest]:

[more information]


• Monument Valley with craters at sunrise.
• Monument Valley at sunrise.
• Cloudy Monument Valley.
• Artistic view of Monument Valley.
• Artistic view of Monument Valley.
• Monument Valley at sunset.
• Monument Valley (bird's-eye view).
• Waltzing Monument Valley (bird's-eye view).
• Waltzing Monument Valley.
• Erosion in Monument Valley (bird's-eye view).
• Erosion in Monument Valley.
• Erosion in Monument Valley with fog.



• The Gate.
• Inside the Gate.
• Beyond the Gate.



Fractal Euclidian Landscapes [Paysages Fractals et Euclidiens]:

• Mountains with an unnatural star-like structure (bird's-eye view).
• Interpolation between a fractal field and a gaussian field (tridimensional visualization).
• Dynamics of archimedian mountains (bird's-eye view).
• Archimedian mountains (bird's-eye view).



• Archimedes mountains (bird's-eye view).
• Foggy Archimedes mountains.
• Soft Archimedes mountains (bird's-eye view).
• Foggy soft Archimedes mountains.



• Archimedes mountains (bird's-eye view).
• Foggy Archimedes mountains.
• Soft Archimedes mountains (bird's-eye view).
• Foggy soft Archimedes mountains.



• Foggy Archimedes mountains.
• Archimedes mountains (bird's-eye view).
• Foggy soft Archimedes mountains.
• Soft Archimedes mountains (bird's-eye view).



Landscape Interpolation [Interpolation de Paysages]:

• Linear landscape interpolation.
• Linear landscape interpolation with fog.



Deep Space [Au plus Profond de l'Espace]:

• More and more craters on the Moon.
• More and more craters on the Moon.
• The surface of the Moon.
• Botticelli anomaly on the Moon.
• Artistic view of the surface of the Moon.
• Alien spacecrafts on the Moon.
• First clouds on the Moon.
• The City and the Stars (a Tribute to Arthur C. Clarke).



Some of my first fractal pictures made in 1980 -very low resolution: 256x256, 8 colors- [Quelques une de mes premières images fractales faites en 1980 -très basse résolution: 256x256, 8 couleurs-]:

• Shore.
• Shore.



• Planet.
• Boy planet.
• Boy planet with a sea.



Tridimensional Structures [Structures Tridimensionnelles]:

• Self-similar zoom in on tridimensional fractal structure.



• Synthesis of tridimensional textures by means of a fractal process.
• Synthesis of tridimensional textures by means of a fractal process.
• Synthesis of tridimensional textures by means of a fractal process.
• Synthesis of tridimensional textures by means of a fractal process.
• Filament dynamics.



• Tridimensional fractal structure.
• Tridimensional fractal structure.
• An interpolation between two fractal networks.



• Dynamics of a bidimensional fractal structure.
• Rotation of a static tridimensional fractal structure.
• Tridimensional fractal structure.



• Tridimensional fractal structure.
• Tridimensional fractal structure.
• Tridimensional fractal structure.
• Tridimensional fractal structure in a tridimensional torus.
• Tridimensional fractal structure in a tridimensional torus.
• Translation along the three axes of a tridimensional fractal structure in a tridimensional torus.
• Tridimensional heterogeneous fractal structure in a tridimensional torus.
• Tridimensional heterogeneous fractal structure in a tridimensional torus.



• Tridimensional fractal structure.
• Tridimensional fractal structure.
• Tridimensional fractal structure.



Bidimensional Erosion Process according to the Sapoval-Baldassarri-Gabrielli Theory [Processus d'Erosion Bidimensionnel suivant la Théorie de Sapoval-Baldassarri-Gabrielli]:

• Foggy cliffs.
• Cliffs at sunset.



• Foggy cliffs.
• Cliffs at sunset.



• Foggy seaboard.
• Foggy 'anti-seaboard'.



• The foggy Earth planet.
• The foggy Earth 'anti-planet'.



• Foggy Brittany.
• Foggy 'anti-Brittany'.



• Foggy island.
• Foggy 'anti-island'.
• Foggy steep island.
• Steep island at sunset.



• Foggy island.
• Foggy 'anti-island'.
• Foggy steep island.
• Steep island at sunset.



• Foggy island.
• Foggy 'anti-island'.
• Foggy steep island.
• Steep island at sunset.



Bidimensional Etching Process according to the Sapoval-Kolwankar Theory [Processus d'Attaque Chimique Bidimensionnel suivant la Théorie de Sapoval-Kolwankar]:

• Foggy cliffs.



Tridimensional Erosion Process according to the Sapoval-Baldassarri-Colonna-Gabrielli Theory [Processus d'Erosion Tridimensionnel suivant la Théorie de Sapoval-Baldassarri-Colonna-Gabrielli]:

• The bottom of the sea.



Diffusion Processes [Processus de Diffusion]:

• Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles.
• Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles with spatial Fourier filtering.
• Bidimensional fractal aggregates obtained by means of two families of interacting particles with the following initial conditions about velocity: V for the Red ones and V/5 for the Blue ones.



• Tridimensional fractal aggregate with variable fractal dimension -increasing from front to back-.



Bidimensional Brownian Motion [Mouvement Brownien Bidimensionnel]:

• Brownian motion of a few heavy slow particles inside a gaz of fast light particles.
• Brownian motion of a few heavy slow particles inside a gaz of fast light particles.
• Brownian motion of a few heavy slow particles inside a gaz of fast light particles with increasing mass ratio (heavy/light=1,10,100,1000).



• Bidimensional brownian motion -dark green- and its 'external border' -white-.
• Bidimensional brownian motion -the colors used (magenta,red,yellow,gree,cyan) are an increasing function of the time- and its 'external border' -white-.



Tridimensional Brownian Motion [Mouvement Brownien Tridimensionnel]:

• Tridimensional brownian motion -dark green- and the 'external border' of its bidimensional projection -white-.
• Tridimensional brownian motion -the colors used (magenta,red,yellow,gree,cyan) are an increasing function of the time- and the 'external border' of its bidimensional projection -white-.



Bidimensional Diffusion Limited Aggregations and Fractal Aggregates [DLAs et Agrégats Fractals Bidimensionnels]:

• Bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.
• The trajectories of bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.
• Bidimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.
• Bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.
• The trajectories of bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.
• Bidimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.



• Bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.
• The trajectories of bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.
• Bidimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.
• Bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.
• The trajectories of bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.
• Bidimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.



Fractal Diffusion Front in a Bidimensional Medium [Front Fractal de Diffusion dans un Milieu Bidimensionnel]:

• Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.
• Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.
• Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.
• Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.



• Tridimensional display (bird's-eye view) of the density of particles during a bidimensional diffusion process.
• Tridimensional display (bird's-eye view) of the density of particles during a bidimensional diffusion process.



• Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles.
• Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles with spatial Fourier filtering.



Fractal Diffusion Front in a Tridimensional Medium [Front Fractal de Diffusion dans un Milieu Tridimensionnel]:

• A gradient of 114299 particles obtains by means of a tridimensional diffusion process.
• A gradient of 128312 particles obtains by means of a tridimensional diffusion process.



Tridimensional Diffusion Limited Aggregations and Fractal Aggregates [DLAs et Agrégats Fractals Tridimensionnels]:

• Tridimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.
• Tridimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.
• Tridimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.
• Tridimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.



• Rotation about the Y (vertical) axis of a tridimensional fractal aggregate that can also be viewed as a set of 4x3 pseudo-stereograms.



• Tridimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.
• The trajectories of tridimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.
• Tridimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.
• Tridimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.
• The trajectories of tridimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.
• Tridimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.



Iterated Function Systems (IFS) [Systèmes de Fonctions Itérées (IFS)]:

• A bidimensional fern computed by means of an 'Iterated Function System' -IFS-.



• A bidimensional Sierpinski carpet computed by means of an 'Iterated Function System' -IFS-.



• A tridimensional Sierpinski 'carpet' computed by means of an 'Iterated Function System' -IFS-.



Recursive "Living" Structures [Structures "Vivantes" Récursives]:

[more information about the Recursive "Living" Structures -in french-]


• Cauliflowers, seaweeds, shells,... with fog.



• Animation of a natural form.
• Alien.
• Alien.
• Animation of a natural form.
• Animation of a natural form.

Copyright (c) Jean-François Colonna, 2000-2010.
Copyright (c) France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) / Ecole Polytechnique, 2000-2010.