Gallery :

Number Theory (and more...)

[Galerie : Théorie des Nombres (et autres sujets...)]

Jean-François COLONNA

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

http://www.lactamme.polytechnique.fr

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(CMAP28 WWW site : this page was created on 03/15/2000 and last updated on 05/16/2012 16:35:59 -CEST-)

• Conway's Surreal Numbers [Les Nombres Surréels de Conway] :
• Space Filling Curve [Courbes remplissant l'Espace] :
• Golden Ratio and Penrose Tilings [Le Nombre d'Or et les pavages de Penrose] :
• Functions of a Complex Variable [Fonctions d'une Variable Complexe] :
• Gamma Function [La Fonction Gamma] :
• Riemann's Zeta Function [La Fonction Zêta de Riemann] :
• Prime Numbers [Les Nombres Premiers] :
• Arrangement of N Points on a Sphere [Placement de N Points sur une Sphère] :
• Knots [Nœuds] :
• Monodimensional Binary Cellular Automata [Automates Cellulaires Binaires Monodimensionnels] :
• Monodimensional "Quasi-Continous" Cellular Automata [Automates Cellulaires "Quasi-Continus" Monodimensionnels] :
• Bidimensional Cellular Automata -the John Conway's bidimensional life game an more- [Automates Cellulaires Bidimensionnels -le jeu de la vie bidimensionnel de John Conway et plus-] :
• Tridimensional Cellular Automata -the John Conway's tridimensional life game an more- [Automates Cellulaires Tridimensionnels -le jeu de la vie Tridimensionnel de John Conway et plus-] :
• Voronoi Diagrams [Diagrammes de Voronoï] :
• Conformal Transformations [Transformations Conformes] :
• Syracuse Sequence [La Suite de Syracuse] :
• Miscellaneous [Divers] :

Number Theory (and more...) [Théorie des Nombres (et autres sujets...)] :

Conway's Surreal Numbers [Les Nombres Surréels de Conway] :

Generation of the 63 first Conway's surreal numbers.

Generation of the 63x63 first Conway's surreal complex numbers.

Generation of the 2x63-1 first Conway's surreal complex numbers.

Generation of the 63x63 first Conway's surreal complex numbers.




Space Filling Curve [Courbes remplissant l'Espace] :

Bidimensional Hilbert Curve -iteration 4-.

Tridimensional Hilbert Curve -iteration 3-.




Golden Ratio and Penrose Tilings [Le Nombre d'Or et les pavages de Penrose] :

The Golden Rectangle.

The two subdivisions of a Golden Triangle.	The two subdivisions of a Golden Triangle.	A random triangular tiling of the plane.	A pseudo-periodical triangular tiling of the plane.	A pseudo-periodical Penrose tiling of the plane.
A pseudo-periodical Penrose tiling of the plane.	A pseudo-periodical non linear Penrose tiling of the plane.	A pseudo-periodical non linear Penrose tiling of the Golden Pentagon.	A tridimensional distorded pseudo-periodical non linear Penrose tiling of the Golden Pentagon.




Functions of a Complex Variable [Fonctions d'une Variable Complexe] :

The relief -modulus- of the function exp(1/z) defined by means of three bidimensional fields.

The relief -modulus- of the function exp(1/z) with argument mapping (bird's-eye view).

The relief -modulus- of the function exp(1/z) with argument mapping.

The relief -modulus- of the function sin(z) defined by means of three bidimensional fields.

The relief -modulus- of the function sin(z) with argument mapping.




Gamma Function [La Fonction Gamma] :

Tridimensional display of the Gamma function inside (-20.0,+20.0)x(-20.0,+20.0) (bird's-eye view).

Tridimensional display of the Gamma function inside (-20.0,+20.0)x(-20.0,+20.0).

Bidimensional zoom in on the Z=Gamma(Z) iteration with display of the arguments.

Tridimensional display of the Z=Gamma(Z) iteration inside (-20.0,+20.0)x(-20.0,+20.0) (bird's-eye view).

Tridimensional display of the Z=Gamma(Z) iteration inside (-20.0,+20.0)x(-20.0,+20.0).




Riemann's Zeta Function [La Fonction Zêta de Riemann] :

Tridimensional display of the Riemann Zeta function inside (+0.1,+0.9)x(0,+50).	Tridimensional display of the Riemann Zeta function inside (-10.0,+20.0)x(-15.0,+15.0) (bird's-eye view).	Tridimensional display of the Riemann Zeta function inside (-10.0,+20.0)x(-15.0,+15.0).	Tridimensional display of the Riemann Zeta function inside (-10.0,+60.0)x(-35.0,+35.0) (bird's-eye view).	Tridimensional display of the Riemann Zeta function inside (-10.0,+60.0)x(-35.0,+35.0).
Tridimensional display of the Riemann Zeta function inside (-50.0,+50.0)x(-50.0,+50.0) (bird's-eye view).	Tridimensional display of the Riemann Zeta function inside (-50.0,+50.0)x(-50.0,+50.0).

Bidimensional zoom in on the Z=Zeta(Z) iteration with display of the arguments.

Tridimensional display of the Z=Zeta(Z) iteration inside (-20.0,+20.0)x(-20.0,+20.0) (bird's-eye view).

Tridimensional display of the Z=Zeta(Z) iteration inside (-20.0,+20.0)x(-20.0,+20.0).

Artistic view of Ph(Zeta).




Prime Numbers [Les Nombres Premiers] :

The Liouville function displayed as a bidimensional random walk for the integer numbers from 2 to 1001.

The Liouville function displayed as a bidimensional random walk for the integer numbers from 2 to 100001.

The Liouville function displayed as a tridimensional random walk for the integer numbers from 2 to 150001.

The special Liouville function displayed as a bidimensional random walk for the integer numbers from 2 to 200001 (Red), 2 to 400001 (Green) and 2 to 800001 (Blue).

The Eratosthene sieve displaying 10x10 numbers.	The Eratosthene sieve displaying 100x100 numbers.	The Ulam spiral displaying 2025 numbers.	The generalized Ulam spiral displaying 81 numbers.	The generalized Ulam spiral displaying 100 numbers.
The generalized Ulam spiral displaying 1024 numbers.	Tridimensional display of the generalized Ulam spiral displaying 4096 numbers.	The Archimedes spiral displaying 100 numbers.	The Archimedes spiral displaying 1000 numbers.	Artistic view of the prime numbers.




Arrangement of N Points on a Sphere [Placement de N Points sur une Sphère] :

'Harmonious' arrangement of 4 points on a sphere -a Tetrahedron- by means of simulated annealing.	'Harmonious' arrangement of 4 points on a sphere -a Tetrahedron- by means of simulated annealing.	'Harmonious' arrangement of 6 points on a sphere -an Octahedron- by means of simulated annealing.	'Harmonious' arrangement of 6 points on a sphere -an Octahedron- by means of simulated annealing.	'Harmonious' arrangement of 8 points on a sphere by means of simulated annealing.
'Harmonious' arrangement of 8 points on a sphere by means of simulated annealing.	'Harmonious' arrangement of 12 points on a sphere -an Icosahedron- by means of simulated annealing.	'Harmonious' arrangement of 12 points on a sphere -an Icosahedron- by means of simulated annealing.	'Harmonious' arrangement of 24 points on a sphere by means of simulated annealing.	'Harmonious' arrangement of 24 points on a sphere by means of simulated annealing.
'Harmonious' arrangement of 26 points on a sphere by means of simulated annealing.




Knots [Nœuds] :

1-foil torus knot on its torus and its asociated Möbius strip.

3-foil torus knot on its torus.

3-foil knot.

5-foil torus knot on its torus.

7-foil torus knot on its torus.




Monodimensional Binary Cellular Automata [Automates Cellulaires Binaires Monodimensionnels ] :

An elementary monodimensional binary cellular automaton -86- with 1 white starting point -bottom middle-.	An elementary monodimensional binary cellular automaton -86- with random white starting points -on the bottom line-.	An elementary monodimensional binary cellular automaton -90- with 1 white starting point -bottom middle-.	An elementary monodimensional binary cellular automaton -90- with 49 white starting points -on the bottom line-.	An elementary monodimensional binary cellular automaton -106- with 1 white starting point -bottom middle-.
An elementary monodimensional binary cellular automaton -106- with random white starting points -on the bottom line-.	An elementary monodimensional binary cellular automaton -110- with 1 white starting point -bottom right-.	An elementary monodimensional binary cellular automaton -110- with 49 white starting points -on the bottom line-.	An elementary monodimensional binary cellular automaton -184- with 1 white starting point -bottom middle-.	An elementary monodimensional binary cellular automaton -184- with random white starting points -on the bottom line-.
Three successive elementary monodimensional binary cellular automata -106,90,86- with 1 yellow starting point -bottom middle-.	Three successive elementary monodimensional binary cellular automata -106,90,86- with random yellow starting points -on the bottom line-.	The alternative use of two elementary monodimensional binary cellular automata -168,165- with 1 white starting point -bottom middle-.	The alternative use of two elementary monodimensional binary cellular automata -168,165- with periodical white starting points -on the bottom line-.	The alternative use of two elementary monodimensional binary cellular automata -168,165- with random white starting points -on the bottom line-.
256 elementary monodimensional binary cellular automata with 1 white starting point (bottom left is the automata number 0 when top right is the number 255).	256 elementary monodimensional binary cellular automata with 56% of white starting points (bottom left is the automata number 0 when top right is the number 255).




Monodimensional "Quasi-Continous" Cellular Automata [Automates Cellulaires "Quasi-Continus" Monodimensionnels] :

A monodimensional 'quasi-continuous' cellular automaton.	A monodimensional 'quasi-continuous' cellular automaton.	A monodimensional 'quasi-continuous' cellular automaton.	A monodimensional 'quasi-continuous' cellular automaton.	A monodimensional 'quasi-continuous' cellular automaton.
A monodimensional 'quasi-continuous' cellular automaton.	A monodimensional 'quasi-continuous' cellular automaton.	Two alternative monodimensional 'quasi-continuous' cellular automata.	Two alternative monodimensional 'quasi-continuous' cellular automata.	A monodimensional 'quasi-continuous' cellular automaton with random perturbation of the rules.
A monodimensional 'quasi-continuous' cellular automaton with random perturbation of the rules.




Bidimensional Cellular Automata -the bidimensional John Conway's life game an more- [Automates Cellulaires Bidimensionnels -le jeu de la vie bidimensionnel de John Conway et plus-] :

The bidimensional John Conway's life game.

Tridimensional display of the dynamics of the bidimensional John Conway's life game.

Tridimensional display of the dynamics of the bidimensional John Conway's life game.

64 elementary bidimensional binary cellular automata with 1 white starting central point.

Tridimensional display of the evolution of a bidimensional binary cellular automaton with 1 white starting central point.




Tridimensional Cellular Automata -the tridimensional John Conway's life game an more- [Automates Cellulaires Tridimensionnels -le jeu de la vie tridimensionnel de John Conway et plus-] :

The tridimensional John Conway's life game with random initial conditions -23% of occupied cells-.	The tridimensional John Conway's life game with random initial conditions -23% of occupied cells-.	The tridimensional John Conway's life game with random initial conditions -24% of occupied cells-.	The tridimensional John Conway's life game with random initial conditions -24% of occupied cells-.	The tridimensional John Conway's life game with random initial conditions -25% of occupied cells-.
The tridimensional John Conway's life game with random initial conditions -25% of occupied cells-.

The tridimensional John Conway's life game with random initial conditions -1.6% of occupied cells-.




Voronoi Diagrams [Diagrammes de Voronoï] :

Voronoi diagrams.




Conformal Transformations [Transformations Conformes] :

1/Z conformal transformation of concentric circles in the complex plane.

ZxZ conformal transformation of concentric circles in the complex plane.

Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/Z conformal transformation in the complex plane.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/Z conformal transformation in the complex plane.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space.
Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a OxO conformal transformation in the pseudo-octonionic space.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a OxO conformal transformation in the pseudo-octonionic space.




Syracuse Sequence [La Suite de Syracuse] :

Syracuse sequence.




Miscellaneous [Divers] :

Table mountain.

Table mountain.

Table mountain.

Table mountain.

Copyright (c) Jean-François Colonna, 2000-2012.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2000-2012.