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Number Theory and much more (Hyperbolic Geometry, Complex Functions, Great Conjectures, Knots, Cellular Automata, Tilings, Meshings, Trees, Fractals,...)

[Galerie (933 images différentes) : Théorie des Nombres et beaucoup d'autres sujets (Géométrie Hyperbolique, Grandes Conjectures, Nœuds, Automates Cellulaires, Pavages, Maillages, Arbres, Fractales,...)]

Jean-François COLONNA
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www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
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Number Theory (and much more...) [Théorie des Nombres (et beaucoup d'autres sujets...)]:

Rational Numbers [Nombres Rationnels]	Real Numbers and Random Walks [Nombres Réels et Marches Aléatoires] Most Important Transcendent Numbers [Les nombres transcendants les plus importants] π [π] The Champernowne Number [Le nombre de Champernowne] Miscellaneous Numbers [Nombres Divers]	Real Numbers, Spirals, Helix and more... [Nombres Réels, Spirales, Hélices et plus...]	Conway's Surreal Numbers [Les Nombres Surréels de Conway]	Extended Arithmetic [Arithmétique étendue]
Space Filling Curves [Courbes remplissant l'Espace] Bidimensional [Bidimensionnelles] Tridimensional [Tridimensionnelles]	Fractal Sets [Ensembles Fractals]	Golden Ratio and Penrose Tilings [Le Nombre d'Or et les Pavages de Penrose]	Euclidian Geometry [Géométrie Euclidienne]	Spherical Geometry [Géométrie Sphérique]
Hyperbolic Geometry [Géométrie Hyperbolique]	Functions of a Complex Variable [Fonctions d'une Variable Complexe]	Functions of an Octonionic Variable [Fonctions d'une Variable Octonionique]	Gamma Function [La Fonction Gamma]	Riemann's Zeta Function [La Fonction Zêta de Riemann]
Prime Numbers [Les Nombres Premiers]	The Continuum Hypothesis [L'Hypothèse du Continu]	The Goldbach Conjecture [La Conjecture de Goldbach]	The ABC Conjecture [La Conjecture ABC]	The Syracuse Conjecture [La Conjecture de Syracuse]
The Additive and Multiplicative Persistence Conjectures [Les Conjectures de Persistance Additive et Multiplicative]	Elliptic Curves [Les Courbes Elliptiques]	Distribution of N Points on a Sphere [Répartition de N Points sur une Sphère]	Point Localization [Localisation de Points]	Bi- and tridimensional Reflections [Réflexions bi- et tridimensionnelles]
Knots [Nœuds]	Monodimensional Binary Cellular Automata [Automates Cellulaires Binaires Monodimensionnels]	Monodimensional "Quasi-Continuous" Cellular Automata [Automates Cellulaires "Quasi-Continus" Monodimensionnels]	Bidimensional Cellular Automata -the John Conway's bidimensional life game and 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 and more- [Automates Cellulaires Tridimensionnels -le Jeu de la Vie Tridimensionnel de John Conway et plus-]
Voronoi Diagrams [Diagrammes de Voronoï]	Conformal Transformations [Transformations Conformes]	Polygons and more [Polygones et plus]	Tilings [Pavages]	Meshings [Maillages]
Deterministic Trees [Arbres déterministes]	Non Deterministic Trees [Arbres non déterministes]	Quadrangulation [Quadrangulation]	Triangulation [Triangulation]	Curves [Courbes]
Surfaces [Surfaces]	Random Surfaces [Surfaces Aléatoires]	Deterministic Fractal Geometry [Géométrie Fractale Déterministe]	Non Deterministic Fractal Geometry and Natural Phenomenon Synthesis [Géométrie Fractale Non Déterministe et Synthèse de Phénomènes Naturels]	Paradoxes [Paradoxes]
Miscellaneous [Divers]

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

Rational Numbers [Nombres Rationnels]:

Bidimensional display of 7 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 15 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 31 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 63 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 127 Rational Numbers by means of the Stern-Brocot Tree.
Bidimensional display of 255 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 32767 Rational Numbers by means of the Stern-Brocot Tree.

Bidimensional display of 7 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 15 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 31 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 63 Rational Numbers by means of the Stern-Brocot Tree.	Bidimensional display of 127 Rational Numbers by means of the Stern-Brocot Tree.
Bidimensional display of 255 Rational Numbers by means of the Stern-Brocot Tree.

Real Numbers and Random Walks [Nombres Réels et Marches Aléatoires]:

Most Important Transcendent Numbers [Les nombres transcendants les plus importants]:

A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of an arbitrary random number (190496...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of the square root of 2 (414213...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'phi' -the golden ratio- (618033...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'e' (718281...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'pi' (141592...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of an arbitrary random number (190496...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of the square root of 2 (414213...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'phi' -the golden ratio- (618033...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'e' (718281...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'pi' (141592...) -base 10- with 33.333 time steps.
A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'pi-e' (423310...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -cartesian coordinates- defined by means of the 99.999 first decimals of 'pi+e' (859874...) -base 10- with 33.333 time steps.	Some tridimensional pseudo-random walks -cartesian coordinates- defined by means of the 99.999 first decimals of 'pi' (141592...) and 'e' (718281...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of an arbitrary random number (190496...) -base 10- with 50.000 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of the square root of 2 (414213...) -base 10- with 50.000 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'phi' -the golden ratio- (618033...) -base 10- with 50.000 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'e' (718281...) -base 10- with 50.000 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of 'pi' (141592...) -base 10- with 50.000 time steps.
A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 100.000 first decimals of the Champernowne number (1 2 3 4 5 6 7 8 9 10 11 12...) -base 10- with 50.000 time steps.

A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of an arbitrary random number (190496...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of the square root of 2 (414213...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'phi' -the golden ratio- (618033...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'e' (718281...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'pi' (141592...) -base 10- with 33.333 time steps.
A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of the Champernowne number (1 2 3 4 5 6 7 8 9 10 11 12...) -base 10- with 33.333 time steps.

A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of an arbitrary random number (190496...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of the square root of 2 (414213...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'phi' -the golden ratio- (618033...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'e' (718281...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'pi' (141592...) -base 10- with 33.333 time steps.
A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'pi-e' (423310...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of 'pi+e' (859874...) -base 10- with 33.333 time steps.	Tridimensional pseudo-random walks -spherical coordinates- defined by means of the 99.999 first decimals of 'pi' (141592...) and 'e' (718281...) -base 10- with 33.333 time steps.	A tridimensional pseudo-random walk -spherical coordinates- defined by means of the 99.999 first decimals of the Champernowne number (1 2 3 4 5 6 7 8 9 10 11 12...) -base 10- with 33.333 time steps.

π [π]:

A bidimensional pseudo-random walk defined by means of 'pi': 141592... -5.000 decimals, -base 10- with 2.500 time steps.

A bidimensional pseudo-random walk defined by means of 'pi': 141592... -10.000 decimals, -base 10- with 10.000 time steps.

A bidimensional pseudo-random walk defined by means of 'pi': 3.141592... -10.000 digits, -base 10- into 021003... -16.607 digits, -base 4-.

A tridimensional pseudo-random walk defined by means of 'pi': 141592... -2.001 digits, -base 10- into 200303... -3.321 digits, -base 4-.

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -10.000 digits, -base 10- into 021003... -16.607 digits, -base 4-.

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -4.000 digits, -base 10- into 050330... -5.138 digits, -base 6-.

A tridimensional pseudo-random walk defined by means of 'pi': 141592... -10.001 digits, -base 10- into 450315... -12.850 digits, base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -10.000 digits, -base 10- into 050330... -12.849 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -10.000 digits, -base 10- into 050330... -12.849 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -10.000 digits, -base 10- into 050330... -12.849 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -10.001 digits, -base 10- into 450315... -12.850 digits, base 6-.
A tridimensional pseudo-random walk defined by means of 'pi': 141592... -10.001 digits, -base 10- into 450315... -12.850 digits, base 6-.

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -100.000 digits, -base 10- into 050330... -128.508 digits, base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -100.000 digits, -base 10- into 050330... -128.508 digits, base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -100.000 digits, -base 10- into 050330... -128.508 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -100.000 digits, -base 10- into 050330... -128.508 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -100.000 digits, -base 10- into 050330... -128.508 digits, -base 6-.
A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -100.000 digits, -base 10- into 305033... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.
A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.	A tridimensional pseudo-random walk defined by means of 'pi': 141592... -100.001 digits, -base 10- into 303142... -128.509 digits, -base 6-.

The Champernowne Number [Le nombre de Champernowne]:

A bidimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 integer numbers-: 0.1 2 3 4 5 6 7 8 9 10 11 12 13... -36.493 digits, base 10-.	A bidimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 integer numbers-: 0.1 2 3 4 5 6 7 8 9 10 11 12 13... -36.493 digits, base 10-.	A bidimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 prime numbers-: 0.2 3 5 7 11 13 17 19 23 29 31... -51.982 digits, base 10-.	A bidimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 prime numbers-: 0.2 3 5 7 11 13 17 19 23 29 31... -51.982 digits, base 10-.	A tridimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 integer numbers-: 0.1 2 3 4 5 6 7 8 9 10 11 12 13... -113.894 digits, base 10- converted into 146.363 digits 042355... -base 6-.
A tridimensional pseudo-random walk defined by means of the Champernowne number -using all base 6 integer numbers-: 0.1 2 3 4 5 10 11 12 13 14 15 20... -130.644 digits, base 6- converted into 130.643 digits 123451... -base 6-.	A tridimensional pseudo-random walk defined by means of the Champernowne number -using all prime numbers-: 0.2 3 5 7 11 13 17 19 23 29 31... -168.982 digits, base 10- into 122525... -217.156 digits, base 6-.	A bidimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 prime numbers-: 0.2 3 5 7 11 13 17 19 23 29 31... -288.982 digits, base 10-.	A bidimensional pseudo-random walk defined by means of the Champernowne number -using all base 10 integer numbers-: 0.1 2 3 4 5 6 7 8 9 10 11 12 13... -1.166.670 digits, base 10-.

Miscellaneous Numbers [Nombres Divers]:

A tridimensional pseudo-random walk defined by means of the square root of 2: 1.414213... -100.000 digits, -base 10- into 225245... -128.508 digits, base 6-.

A tridimensional pseudo-random walk defined by means of 'phi' -the golden ratio-: 1.618033... -100.000 digits, base 10- into 341254... -128.508 digits, -base 6-.

A tridimensional pseudo-random walk defined by means of 'e': 2.718281... -100.000 digits, base 10- into 415052... -128.508 digits, base 6-.

Real Numbers, Spirals, Helix and more... [Nombres Réels, Spirales, Hélices et plus...]:

The DNA of Mathematics -the 100 first digits of 'pi' and '2.pi'-.

The DNA of Mathematics -the 480 first digits of 'pi' and '2.pi'-.

The DNA of Mathematics -the 60 first digits of 'pi' and '1/pi'-.

The DNA of Mathematics -the 480 first digits of 'pi' and '1/pi'-.

The DNA of Mathematics -the 60 first digits of 'pi' and '1/pi'-.

The DNA of Mathematics -the 480 first digits of 'pi' and '1/pi'-.

The DNA of Mathematics -the 100 first digits of 'pi' and '1/pi'-.

The DNA of Mathematics -the 480 first digits of 'pi' and '1/pi'-.

The DNA of Mathematics -the 60 first digits of 'pi' and 'e'-.

The DNA of Mathematics -the 480 first digits of 'pi' and 'e'-.

The DNA of Mathematics -the 100 first digits of 'pi' and 'e'-.

The DNA of Mathematics -the 480 first digits of 'pi' and 'e'-.

An Archimedes spiral displaying 'pi' with 50 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 50 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 50 digits -base 10-.

An Archimedes spiral displaying 'pi' with 50 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 50 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 50 digits -base 10-.

An Archimedes spiral displaying 'pi' with 50 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 50 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 50 digits -base 10-.

An Archimedes spiral displaying 'pi' with 100 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 100 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 100 digits -base 10-.

An Archimedes spiral displaying 'pi' with 100 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 100 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 100 digits -base 10-.

An Archimedes spiral displaying 'pi' with 200 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 200 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 200 digits -base 10-.

An Archimedes spiral displaying 'pi' with 200 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 200 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 200 digits -base 10-.

An Archimedes spiral displaying 'pi' with 1.000 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 1.000 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 1.000 digits -base 10-.

An Archimedes spiral displaying 'pi' with 1.000 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 1.000 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 1.000 digits -base 10-.

An Archimedes spiral displaying 'pi' with 10.000 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 10.000 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 10.000 digits -base 10-.

An Archimedes spiral displaying 'pi' with 10.000 digits -base 10-.

Tridimensional display -bird's-eye view- of an Archimedes spiral displaying 'pi' with 10.000 digits -base 10-.

Tridimensional display of an Archimedes spiral displaying 'pi' with 10.000 digits -base 10-.

Tridimensional display -bird's-eye view- of a spiral displaying 'pi' with 2.000 digits -base 10-.

Tridimensional display of a spiral displaying 'pi' with 2.000 digits -base 10-.

Tridimensional display of a spiral displaying 'pi' with 4.000 digits -base 10-.

An Archimedes spiral displaying the 100.000 first digits -base 10- of 'pi'.

The 1.000 first digits -base 10- of 'pi' displayed on an helix -bad point of view-.

display of 'pi' with 100 digits {3.141592...} on an helix -grey-.

display of 'pi' with 1.000 digits -base 10- on an helix -good point of view-.

The God Chessboard or the 64 first 'decimals' -base 2- of 'pi' -the first 'decimal' is the bottom left black square-.

The 262.144 first digits -base 2- of 'pi' -the first 'digit' is the top left white square-.

The 65.536 first 'decimals' -base 3- of 'pi' -the first 'decimal' is the top left red square-.

The 262.144 first decimals -base 3- of 'pi' -the first 'digit' is the top left red square-.

The 1.048.576 first 'decimals' -base 3- of 'pi' -the first 'decimal' is the top left red square-.

The 4 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 1-.	The 16 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 2-.	The 64 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 3-.	The 256 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 4-.	The 1.024 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 5-.
The 4.096 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 6-.	The 16.384 first decimals -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 7-.

The 8 first decimals -base 10- of 'pi' on a Tridimensional Hilbert Curve -iteration 1-.

The 64 first decimals -base 10- of 'pi' on a Tridimensional Hilbert Curve -iteration 2-.

The 512 first decimals -base 10- of 'pi' on a Tridimensional Hilbert Curve -iteration 3-.

The 4.096 first decimals -base 10- of 'pi' on a Tridimensional Hilbert Curve -iteration 4-.

The 50 first digits {141592...} of 'pi' displayed on an helix -orange-.

The 50 first digits {7,1,8,2,8,1,8,2,8,4,...} of 'e' displayed on an helix -orange-.

The 50 first digits {4,1,4,2,1,3,5,6,2,3,...} of the square root of 2 displayed on an helix -orange-.

The 50 first digits {6,1,8,0,3,3,9,8,8,7,...} of the golden ratio displayed on an helix -orange-.

The 50 first digits {1,2,3,4,5,6,7,8,9,1,...} of the Champernowne number displayed on an helix -orange-.

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.

Extended Arithmetic [Arithmétique étendue]:

A gaussian field computed with the standard arithmetic (addition,multiplication).

A gaussian field computed with the 'max-plus' arithmetic (maximum,addition).

A gaussian field computed with the 'min-plus' arithmetic (minimum,addition).

A sphere computed with the standard arithmetic (addition,multiplication).

A sphere computed with the 'max-plus' arithmetic (maximum,addition).

A sphere computed with the 'min-plus' arithmetic (minimum,addition).

Fractal Sets [Ensembles Fractals]:

The Cantor Triadic Set -iterations 0 to 5-.	The first four iterations of the construction of the von Koch snowflake.	The Sierpinski Carpet -iteration 1 to 5-.	The Menger Sponge -iteration 5-.	An amazing cross-section inside the Menger Sponge -iteration 5-.
An amazing generalized cross-section inside the Menger Sponge -iteration 5-.	A spherical cross-section inside the Menger Sponge -iteration 5-.

Space Filling Curves [Courbes remplissant l'Espace]:

Bidimensional Space Filling Curves [Courbes Bidimensionnelles remplissant l'Espace]:

The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 2 digits-.

The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 4 digits-.

The [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 6 digits-.

The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 8 digits-.

Bidimensional Hilbert Curve -iterations 1 to 5-.

Bidimensional Hilbert Curve -iteration 1-.

Bidimensional Hilbert Curve -iteration 2-.

Bidimensional Hilbert Curve -iteration 3-.

Bidimensional Hilbert Curve -iteration 4-.

Bidimensional Hilbert Curve -iteration 5-.

Bidimensional Hilbert Curve -iteration 1-.	Bidimensional Hilbert Curve -iteration 2-.	Bidimensional Hilbert Curve -iteration 3-.	Bidimensional Hilbert Curve -iteration 4-.	Bidimensional Hilbert Curve -iteration 5-.
Bidimensional Hilbert Curve -iteration 6-.

A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} -iteration 1-.	A Bidimensional Hilbert-like Curve defined with {X₂(...),Y₂(...)} -iteration 2-.	A Bidimensional Hilbert-like Curve defined with {X₃(...),Y₃(...)} -iteration 3-.	A Bidimensional Hilbert-like Curve defined with {X₄(...),Y₄(...)} -iteration 4-.	A Bidimensional Hilbert-like Curve defined with {X₅(...),Y₅(...)} -iteration 5-.
A Bidimensional Hilbert-like Curve defined with {X₆(...),Y₆(...)} -iteration 6-.

A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} -iteration 1-.	A Bidimensional Hilbert-like Curve defined with {X₂(...),Y₂(...)} -iteration 2-.	A Bidimensional Hilbert-like Curve defined with {X₃(...),Y₃(...)} -iteration 3-.	A Bidimensional Hilbert-like Curve defined with {X₄(...),Y₄(...)} -iteration 4-.	A Bidimensional Hilbert-like Curve defined with {X₅(...),Y₅(...)} -iteration 5-.
A Bidimensional Hilbert-like Curve defined with {X₆(...),Y₆(...)} -iteration 6-.

A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} related to the Mandelbrot set border -iteration 1-.	A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} related to the Mandelbrot set border -iteration 2-.	A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} related to the Mandelbrot set border -iteration 3-.	A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} related to the Mandelbrot set border -iteration 4-.	A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} related to the Mandelbrot set border -iteration 5-.
A Bidimensional Hilbert-like Curve defined with {X₁(...),Y₁(...)} related to the Mandelbrot set border -iteration 6-.

A sphere described by means of a Bidimensional Hilbert Curve -iteration 5-.

A torus described by means of a Bidimensional Hilbert Curve -iteration 5-.

The Bonan-Jeener double bottle described by means of a Bidimensional Hilbert Curve -iteration 5-.

A sphere described by means of a Bidimensional Hilbert Curve -iteration 7-.	A 'crumpled' sphere described by means of a Bidimensional Hilbert Curve -iteration 7-.	A torus described by means of a Bidimensional Hilbert Curve -iteration 7-.	The Bonan-Jeener double bottle described by means of a Bidimensional Hilbert Curve -iteration 7-.	The Möbius strip described by means of a Bidimensional Hilbert Curve -iteration 7-.
The Klein bottle described by means of a Bidimensional Hilbert Curve -iteration 7-.	The Klein bottle described by means of a Bidimensional Hilbert Curve -iteration 7-.	The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 6-.	The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 6-.	The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 7-.
The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-.	The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-.	The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-.

A sphere described by means of a Bidimensional Peano Curve -6 digits-.

A torus described by means of a Bidimensional Peano Curve -6 digits-.

The Bonan-Jeener double bottle described by means of a Bidimensional Peano Curve -6 digits-.

A sphere described by means of a Bidimensional Peano Curve -8 digits-.

A torus described by means of a Bidimensional Peano Curve -8 digits-.

The Bonan-Jeener double bottle described by means of a Bidimensional Peano Curve -8 digits-.

The Möbius strip described by means of a Bidimensional Peano Curve -8 digits-.

The Klein bottle described by means of a Bidimensional Peano Curve -8 digits-.

A sphere described by means of a Bidimensional Hilbert-like Curve -iteration 5-.

A torus described by means of a Bidimensional Hilbert-like Curve -iteration 5-.

The Bonan-Jeener double bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-.

A sphere described by means of a Bidimensional Hilbert-like Curve -iteration 6-.	A torus described by means of a Bidimensional Hilbert-like Curve -iteration 6-.	The Bonan-Jeener double bottle described by means of a Bidimensional Hilbert-like Curve -iteration 6-.	The Möbius strip described by means of a Bidimensional Hilbert-like Curve -iteration 6-.	The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 6-.
The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 6-.

Tridimensional representation of a quadridimensional Calabi-Yau manifold described by means of 5x5 Bidimensional Hilbert Curves -iteration 4-.

Tridimensional representation of a quadridimensional Calabi-Yau manifold described by means of 5x5 Bidimensional Hilbert Curves -iteration 5-.

Tridimensional Space Filling Curves [Courbes Tridimensionnelles remplissant l'Espace]:

Tridimensional Hilbert Curve -iterations 1 to 3-.

Tridimensional Hilbert Curve -iteration 1-.

Tridimensional Hilbert Curve -iteration 2-.

Tridimensional Hilbert Curve -iteration 3-.

Tridimensional Hilbert Curve -iteration 4-.

Tridimensional Hilbert Curve -iteration 1-.

Tridimensional Hilbert Curve -iteration 2-.

Tridimensional Hilbert Curve -iteration 3-.

Tridimensional Hilbert Curve -iteration 4-.

The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 3 digits-.

The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 6 digits-.

The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 9 digits-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 3-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 3-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 3-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 3-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 3-foil torus knot -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 3-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 3-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 3-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 3-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 3-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 3-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 3-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 3-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 3-foil torus knot -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 5-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 5-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 5-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 5-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 5-foil torus knot -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 7-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 7-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 7-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 7-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 7-foil torus knot -iteration 5-.

A parallelepipedic Torus described by means of a Tridimensional Hilbert Curve -iteration 4-.

A Jeener-Möbius Tridimensional manifold described by means of a Tridimensional Hilbert Curve -iteration 4-.

A parallelepipedic Torus described by means of an 'open' 3-foil torus knot -iteration 4-.

A Jeener-Möbius Tridimensional manifold described by means of an 'open' 3-foil torus knot -iteration 4-.

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

The Golden Rectangle.

Recursive subdivision of the Golden Rectangle by means of the Golden Ratio -phi-.

The Golden Rectangle.

Recursive subdivision of four Golden Rectangles -a Tribute to Piet Mondrian-.

The two subdivisions of the 'flat' Golden Triangle.

The 'flat' Golden Triangle.

One of the two subdivisions of the 'flat' Golden Triangle.

The two subdivisions of the 'sharp' Golden Triangle.

The 'sharp' Golden Triangle.

One of the two subdivisions of the 'sharp' Golden Triangle.

The construction process of an aperiodic Penrose tiling -a large golden triangle-.

The construction process of an aperiodic Penrose tiling -1 random subdivision iteration-.

The construction process of an aperiodic Penrose tiling -6 random subdivision iterations-.

The construction process of an aperiodic Penrose tiling -the erasing process-.

The construction process of an aperiodic Penrose tiling -the aperiodic Penrose tiling-.

The parallel axiom of the Euclidian Geometry.

A plane -zero curvature-.

The Spherical Geometry.

An hexagonal tiling of the hyperbolic Poincaré disk -iteration 5-.

A one sheet hyperboloid of revolution -negative curvature-.

A random triangular tiling of the plane.

A pseudo-periodical triangular tiling of the plane.

The erasing of 'common' edges -red-.

An aperiodic Penrose tiling of the plane.

An aperiodic non linear Penrose tiling of the plane.

An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose-.	An aperiodic Penrose tiling of the plane -the extended paradoxal demoniac Rubik's Cube-.	An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose-.
An aperiodic non linear Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.	The Penrose city.
A tridimensionally distorded aperiodic Penrose tiling of the plane.	A tridimensionally distorded aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.

An aperiodic Penrose tiling of the Golden Decagon.

A tridimensionally distorded aperiodic Penrose tiling of the Golden Decagon.

A tridimensionally distorded -by means of a fractal bidimensional height field- aperiodic Penrose tiling of the Golden Decagon.

An aperiodic Penrose tiling of the Golden Decagon with five hidden cubes.

An aperiodic Penrose tiling of the Golden Decagon.

Artistic view of an aperiodic Penrose tiling of the Golden Decagon.

An aperiodic Penrose tiling of the Golden Decagon -a Tribute to Piet Mondrian and Roger Penrose-.

A 'triple stack' of two aperiodic Penrose tilings of the plane.

A tridimensional structure made of six Golden Decagons with aperiodic Penrose tilings.

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).

Functions of an Octonionic Variable [Fonctions d'une Variable Octonionique]:

The 'exponential' of a sphere.	The 'cosine' of a sphere.	The 'sine' of a sphere.	The 'tangent' of a sphere.	The 'hyperbolic cosine' of a sphere.
The 'hyperbolic sine' of a sphere.	The 'hyperbolic tangent' of a sphere.

The 'exponential' of a sphere.	The 'cosine' of a sphere.	The 'sine' of a sphere.	The 'tangent' of a sphere.	The 'hyperbolic cosine' of a sphere.
The 'hyperbolic sine' of a sphere.	The 'hyperbolic tangent' of a sphere.

Prime Numbers [Les Nombres Premiers]:

The Radical of the integer numbers from 2 to 257.

The smooth integers.

The smooth integers: the prime factor sum of the integer numbers.

The smooth integers: the prime factor product -the radical function- of the integer numbers.

The 2-smooth integers.

The 3-smooth integers.

The 5-smooth integers.

The 7-smooth integers.

The 2/3/5/7-smooth integers.

The 2/3/5/7-smooth integers on a generalized Ulam spiral.

The 2/3/5/7/11/13/17-smooth integers on a generalized Ulam spiral.

The K-smooth integers on a generalized Ulam spiral.

The K-smooth integers on a Bidimensional Hilbert Curve -iteration 1-.

The K-smooth integers on a Bidimensional Hilbert Curve -iteration 2-.

The K-smooth integers on a Bidimensional Hilbert Curve -iteration 3-.

The K-smooth integers on a Bidimensional Hilbert Curve -iteration 4-.

The K-smooth integers on a Bidimensional Hilbert Curve -iteration 5-.

The K-smooth integers on a Tridimensional Hilbert Curve -iteration 1-.

The K-smooth integers on a Tridimensional Hilbert Curve -iteration 2-.

The K-smooth integers on a Tridimensional Hilbert Curve -iteration 3-.

The K-smooth integers on a Tridimensional Hilbert Curve -iteration 4-.

The 128 first prime numbers displayed modulo the 16 smallest ones -spiral like trajectories-.

The 8192 first prime numbers displayed modulo the 16 smallest ones -ramdom walk like trajectories-.

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 100001.

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 sum -top, white- of 96 cosine lines -the 12 first, colors- with the 25 first prime numbers -white vertical lines-.

The Eratosthene sieve displaying the integer numbers from 1 to 128.

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 100 numbers.	The generalized Ulam spiral displaying 1024 numbers.
The generalized Ulam spiral displaying 1024 numbers.	The generalized Ulam spiral.

Tridimensional display of the generalized Ulam spiral displaying 4096 numbers.

An Archimedes spiral displaying 100 numbers.

An Archimedes spiral displaying 1000 numbers.

An Archimedes spiral displaying 5000 numbers.

An Archimedes spiral displaying 10000 numbers.

An Archimedes spiral displaying 50000 numbers.

An Archimedes spiral displaying 100000 numbers.

The Ulam spiral.

The Ulam spiral with display of the first twin prime numbers ('2-twin' prime numbers).

The Ulam spiral with display of the first '4-twin' prime numbers.

The Ulam spiral with display of the first '6-twin' prime numbers.

The Ulam spiral with display of the first '8-twin' prime numbers.

The number of divisors of the 4 first integer numbers on a Bidimensional Hilbert Curve -iteration 1-.

The number of divisors of the 16 first integer numbers on a Bidimensional Hilbert Curve -iteration 2-.

The number of divisors of the 64 first integer numbers on a Bidimensional Hilbert Curve -iteration 3-.

The number of divisors of the 256 first integer numbers on a Bidimensional Hilbert Curve -iteration 4-.

The number of divisors of the 1.024 first integer numbers on a Bidimensional Hilbert Curve -iteration 5-.

The prime numbers among the 4 first integer numbers on a Bidimensional Hilbert Curve -iteration 1-.

The prime numbers among the 16 first integer numbers on a Bidimensional Hilbert Curve -iteration 2-.

The prime numbers among the 64 first integer numbers on a Bidimensional Hilbert Curve -iteration 3-.

The prime numbers among the 256 first integer numbers on a Bidimensional Hilbert Curve -iteration 4-.

The prime numbers among the 1024 first integer numbers on a Bidimensional Hilbert Curve -iteration 5-.

The number of divisors of the 8 first integer numbers on a Tridimensional Hilbert Curve -iteration 1-.

The number of divisors of the 64 first integer numbers on a Tridimensional Hilbert Curve -iteration 1-.

The number of divisors of the 512 first integer numbers on a Tridimensional Hilbert Curve -iteration 1-.

The number of divisors of the 4096 first integer numbers on a Tridimensional Hilbert Curve -iteration 1-.

The prime numbers among the 8 first integer numbers on a Tridimensional Hilbert Curve -iteration 1-.

The prime numbers among the 64 first integer numbers on a Tridimensional Hilbert Curve -iteration 2-.

The prime numbers among the 512 first integer numbers on a Tridimensional Hilbert Curve -iteration 3-.

The prime numbers among the 4096 first integer numbers on a Tridimensional Hilbert Curve -iteration 4-.

The distance -an even number- between consecutive prime numbers displayed by means of increasing luminance colors.

Artistic view of the prime numbers.

The Continuum Hypothesis [L'Hypothèse du Continu]:

The four first power sets {P(E),P(P(E)),P(P(P(E))),P(P(P(P(E))))} of a one-element set E.

The Continuum Hypothesis (CH).

The Continuum Hypothesis (CH) -an allegory-.

The Goldbach Conjecture [La Conjecture de Goldbach]:

The Goldbach conjecture.

The Goldbach conjecture -the Goldbach comet- for the even numbers from 6 to 6244.

The Goldbach conjecture -the Goldbach comet or the Goldbach rainbow- for the even numbers from 6 to 41518.

The Goldbach conjecture -the Goldbach comet- for the even numbers from 6 to 411678.

The Goldbach conjecture -the Goldbach comet or the Goldbach rainbow- for the even numbers from 6 to 411678.

The Goldbach conjecture for the even numbers from 6 to 1564.

The Goldbach conjecture -the Goldbach glacier- for the even numbers from 6 to 6244.

Tridimenional display of the Goldbach conjecture.

The ABC Conjecture [La Conjecture ABC]:

The Radical of the integer numbers from 2 to 257.

The ABC conjecture.

Tridimensional visualization of the ABC conjecture.

The Syracuse Conjecture [La Conjecture de Syracuse]:

The Syracuse conjecture for U(0)={1,2,3,4,...,262144}.

The Syracuse conjecture for U(0)={2,4,6,8,...,524288}.

The Syracuse conjecture for U(0)={3,6,9,12,...,786432}.

The Syracuse conjecture for U(0)={5,10,15,20,...,1310720}.

The Syracuse conjecture for U(0)={7,14,21,28,...,1835008}.

The Syracuse conjecture for U(0)={1,2,3,4,...,128} -monodimensional display-.

The Syracuse conjecture for U(0)={1,2,3,4,...,128} -monodimensional display of the parities-.

The Syracuse conjecture for U(0)={1,2,3,4,...,256} -monodimensional display-.

The Syracuse conjecture for U(0)={2,4,6,8,...,256} -monodimensional display-.

The Syracuse conjecture for U(0)={3,6,9,12,...,384} -monodimensional display-.

The Syracuse conjecture for U(0)={5,10,15,20,...,640} -monodimensional display-.

The Syracuse conjecture for U(0)={7,14,21,28,...,896} -monodimensional display-.

The Syracuse conjecture for U(0)={2,3,5,7,...,719} -monodimensional display-.

The Syracuse conjecture for U(0)={1,2,3,4,...,256} -monodimensional display-.

The Syracuse conjecture for U(0)={5,6,7,8,...,20} -bidimensional display-.

The Syracuse conjecture for U(0)={5,6,7,8,...,20} -tridimensional display-.

The Syracuse conjecture for U(0)={5,6,7,8,...,20} -bidimensional display-.

The Syracuse conjecture for U(0)={5,6,7,8,...,20} -tridimensional display-.

The Syracuse conjecture for U(0)={5,6,7,8,...,20} -polar coordinates display-.

The Syracuse conjecture for U(0)={5,6,7,8,...,68} -'pseudo-ramdom walk' display: 'The Syracuse Sun'-.

The Syracuse conjecture for U(0)={5,6,7,8,...,68} -'pseudo-ramdom walk' display: 'The Syracuse Pulsar'-.

The Syracuse conjecture for U(0)={5,6,7,8,...,260} -'pseudo-ramdom walk' display: 'The Syracuse Supernova'-.

The Additive and Multiplicative Persistence Conjectures [Les Conjectures de Persistance Additive et Multiplicative]:

The additive persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right-.

The multiplicative persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right-.

Elliptic Curves [Les Courbes Elliptiques]:

The abelian -commutative- group defined on elliptic curves.

Distribution of N Points on a Sphere [Répartition de N Points sur une Sphère]:

4 evenly distributed points on a sphere -a Tetrahedron- by means of simulated annealing.

6 evenly distributed points on a sphere -an Octahedron- by means of simulated annealing.

8 evenly distributed points on a sphere by means of simulated annealing.

12 evenly distributed points on a sphere -an Icosahedron- by means of simulated annealing.

24 evenly distributed points on a sphere by means of simulated annealing.

26 evenly distributed points on a sphere by means of simulated annealing.

2000 evenly distributed points on a sphere by means of the Fibonacci spiral.

4 evenly distributed points on a sphere -a Tetrahedron- by means of simulated annealing.

4 distributed points on a sphere by means of the Fibonacci spiral.

6 evenly distributed points on a sphere -an Octahedron- by means of simulated annealing.

6 distributed points on a sphere by means of the Fibonacci spiral.

8 evenly distributed points on a sphere by means of simulated annealing.

8 distributed points on a sphere by means of the Fibonacci spiral.

12 evenly distributed points on a sphere -an Icosahedron- by means of simulated annealing.

12 distributed points on a sphere by means of the Fibonacci spiral.

20 evenly distributed points on a sphere by means of simulated annealing.

20 distributed points on a sphere by means of the Fibonacci spiral.

24 evenly distributed points on a sphere by means of simulated annealing.

24 distributed points on a sphere by means of the Fibonacci spiral.

Point Localization [Localisation de Points]:

Bidimensional localization of a point P its distances to the three vertices of a triangle ABC being known, the four points being coplanar.

Localization of a point P its distances to the four vertices of a tetrahedron ABCD being known.

Tridimensional localization of a point P its distances to the four vertices of a tetrahedron ABCD being known.

Bi- and tridimensional Reflections [Réflexions bi- et tridimensionnelles]:

The 'green' reflection of a red triangle obtained by a 'blue' symmetry of each red vertex about the opposite red side.

The double reflection -green- of a small arbitrary triangle -center red-.

The double reflection -green- of a small equilateral triangle -center red-.

The Generalization of the reflection of a triangle.

The 'green' reflection of a red tetrahedron obtained by a 'blue' symmetry of each red vertex about the opposite red face.

The double reflection -green- of a small arbitrary tetrahedron -center red-.

The double reflection -green- of a small regular tetrahedron -center red-.

Knots [Nœuds]:

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

3-foil knot.

3-foil torus knot on its torus.

5-foil torus knot on its torus.

7-foil torus knot on its torus.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 3-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 3-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 3-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 3-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 3-foil torus knot -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 3-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 3-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 3-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 3-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 3-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 3-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 3-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 3-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 3-foil torus knot -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 5-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 5-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 5-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 5-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 5-foil torus knot -iteration 5-.

A Tridimensional Hilbert-like Curve defined with {X₁(...),Y₁(...),Z₁(...)} and based on an 'open' 7-foil torus knot -iteration 1-.

A Tridimensional Hilbert-like Curve defined with {X₂(...),Y₂(...),Z₂(...)} and based on an 'open' 7-foil torus knot -iteration 2-.

A Tridimensional Hilbert-like Curve defined with {X₃(...),Y₃(...),Z₃(...)} and based on an 'open' 7-foil torus knot -iteration 3-.

A Tridimensional Hilbert-like Curve defined with {X₄(...),Y₄(...),Z₄(...)} and based on an 'open' 7-foil torus knot -iteration 4-.

A Tridimensional Hilbert-like Curve defined with {X₅(...),Y₅(...),Z₅(...)} and based on an 'open' 7-foil torus knot -iteration 5-.

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 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-Continuous" 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.

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

Bidimensional Cellular Automata -the bidimensional John Conway's life game and 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.

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 and 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-.

The tridimensional John Conway's life game with random initial conditions -0.4% of occupied cells- and 40 iterations.

The tridimensional John Conway's life game with random initial conditions -0.8% of occupied cells- and 40 iterations.

The tridimensional John Conway's life game with random initial conditions -1.2% of occupied cells- and 40 iterations.

The tridimensional John Conway's life game with random initial conditions -1.6% of occupied cells- and 40 iterations.

The tridimensional John Conway's life game with random initial conditions -0.3% of occupied cells- and 10 iterations.

The tridimensional John Conway's life game with random initial conditions -0.3% of occupied cells- and 20 iterations.

The tridimensional John Conway's life game with random initial conditions -0.3% of occupied cells- and 30 iterations.

The tridimensional John Conway's life game with random initial conditions -0.3% of occupied cells- and 40 iterations.

Voronoi Diagrams [Diagrammes de Voronoï]:

Voronoi diagrams.

Untitled 0614.

Conformal Transformations [Transformations Conformes]:

Z² conformal transformation of concentric circles in the complex plane.

1/Z 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 -tridimensional cross-section-.

Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.
A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.

Close-up on a foggy pseudo-quaternionic Mandelbrot set with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a foggy pseudo-quaternionic Mandelbrot set with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a foggy pseudo-quaternionic Mandelbrot set with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a foggy pseudo-quaternionic Mandelbrot set with a (1/O)² conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.
Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.

Close-up on a foggy pseudo-octonionic Mandelbrot set with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a foggy pseudo-octonionic Mandelbrot set with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a foggy pseudo-octonionic Mandelbrot set with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a foggy pseudo-quaternionic Mandelbrot set with a (1/O)³ conformal transformation in the octonionic space -tridimensional cross-section-.
Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Tridimensional visualization of the Verhulst dynamics with a Q² conformal transformation in the pseudo-quaternionic space.	Tridimensional visualization of the Verhulst dynamics with a Q² conformal transformation in the pseudo-quaternionic space.	Tridimensional visualization of the Verhulst dynamics with a Q² conformal transformation in the pseudo-quaternionic space.	Tridimensional visualization of the Verhulst dynamics with a 1/Q conformal transformation in the pseudo-quaternionic space.	Tridimensional visualization of the Verhulst dynamics with a 1/Q conformal transformation in the pseudo-quaternionic space.
Tridimensional visualization of the Verhulst dynamics with a 1/Q conformal transformation in the pseudo-quaternionic space.	Tridimensional visualization of the Verhulst dynamics with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a O² conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a O² conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a O² conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a O² conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.	Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a O² conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.
A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a O² conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.

A tridimensional cubic mesh with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

A tridimensional cubic mesh with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

A truncated tridimensional sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

A truncated tridimensional sphere with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

A truncated tridimensional torus with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

The Jeener-Klein quintuple bottle with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

A truncated quadrimensional Calabi-Yau manifold with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	'Fractal set of ropes' based on a plane with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	'Fractal set of ropes' based on a plane with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Fractal 'celestial body' based on a torus with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.
Fractal 'celestial body' based on a torus with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	Fractal 'celestial body' based on the Bonan-Jeener-Klein triple bottle with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

A tridimensional structure based on a 1/O conformal transformation in the pseudo-octonionic space -tridimensional cross-section-.

A pseudo-quaternionic Julia set ('MandelBulb' like: a 'JuliaBulb') computed with A=(-0.581514...,+0.635888...,0,0) and with a rotation about the Y axis and with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Detail close-up of the quaternionic Julia set -degree=2- computed -pseudo-addition and pseudo-multiplication in C- with A=(-0.581514...,+0.635888...,0,0) and with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Zoom in on a pseudo-octonionic Julia set ('MandelBulb' like: a 'JuliaBulb') computed with A=(-0.581514...,+0.635888...,0,0,0,0,0,0) with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

An amazing cross-section inside the Menger Sponge -iteration 4- with a O² conformal transformation in the octonionic space -tridimensional cross-section-.

An amazing cross-section inside the Menger Sponge -iteration 5- with a O² conformal transformation in the octonionic space -tridimensional cross-section-.

An amazing cross-section inside the Menger Sponge -iteration 4- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	An amazing cross-section inside the Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	An amazing cross-section inside the Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	The Menger Sponge -iteration 1- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	The Menger Sponge -iteration 2- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.
The Menger Sponge -iteration 3- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	The Menger Sponge -iteration 4- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.	The Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

An amazing cross-section inside the Menger Sponge -iteration 4- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An amazing cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An amazing cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An amazing generalized cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	The Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.
A spherical cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.
An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	A full-random extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	A half-random extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	A half-random extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.
A parallelepipedic extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.
An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	An amazing cross-section inside an extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	A 'pyramidal' cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-, 'Ô temps tes pyramides' -a Tribute to Jorge Luis Borges-.	A 'double conic' cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.	A 'conic' cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.
The intersection of the Menger Sponge -iteration 5- and of a quadridimensional Calabi-Yau manifold -tridimensional representation- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

The erosion of an amazing cross-section inside the Menger Sponge -iteration 1- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

The erosion of an amazing cross-section inside the Menger Sponge -iteration 2- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

The erosion of an amazing cross-section inside the Menger Sponge -iteration 3- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

The erosion of an amazing cross-section inside the Menger Sponge -iteration 4- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

The erosion of an amazing cross-section inside the Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

A tridimensional pseudo-random walk defined by means of the 8.188 first prime numbers starting at 11 computed modulo 7 minus 1: 352410... -8.188 digits, -base 6- into 352410... -8.188 digits, -base 6- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Tridimensional fractal structure with a O² conformal transformation in the octonionic space -tridimensional cross-section-.

Tridimensional fractal structure with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Tridimensional fractal structure with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Tridimensional 'smooth' fractal surface with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

Polygons and more [Polygones et plus]:

A regular 3-gon -an equilateral triangle-.	A regular 4-gon -a square-.	A regular 5-gon -a pentagon-.	A regular 6-gon -an hexagon-.	A regular 7-gon -an heptagon-.
A regular 8-gon -an octogon-.

An equilateral triangle.

A right-angled triangle.

A demonstration of the Pythagoras' theorem.

The pythagorician knotted rope.

A circle.

Tilings [Pavages]:

A periodical tiling of the plane using isosceles right angled triangles.

A periodical tiling of the plane using squares.

A periodical tiling of the plane using equilateral triangles.

A periodical tiling of the plane using hexagons.

A periodical tiling of the plane using isosceles right angled triangles.

A periodical tiling of the plane using squares.

A periodical tiling of the plane using equilateral triangles.

A periodical tiling of the plane using hexagons.

A random tiling of a square domain using dominoes (1x2 rectangles) -line after line-.

A random tiling of a square domain using dominoes (1x2 rectangles) -column after column-.

A random tiling of a square domain using dominoes (1x2 rectangles) -following a 'square' spiral-.

A random tiling of a square domain using dominoes (1x2 rectangles) -following a random order-.

A random tiling of a square domain using dominoes (1x2 rectangles) -line after line- with display of clusters of vertical rectangles using the 4-connexity.

A random tiling of a square domain using dominoes (1x2 rectangles) -column after column- with display of clusters of horizontal rectangles using the 4-connexity.

A random tiling of a square domain using dominoes (1x2 rectangles) -line after line- with display of clusters of horizontal and vertical rectangles using the 4-connexity.

The 'EinStein' aperiodic 'Spectre' tile.

The 'Mystic' made of two 'EinStein' aperiodic 'Spectre' tiles.

The level-1 cluster made of 8 'Spectre' tiles with display of all the key-points making quadrilaterals (7 red small and a green big one).

The level-1 cluster made of 9 'Spectre' tiles with display of all the key-points making quadrilaterals (8 red small and a green big one).

The level-1 cluster made of 8 'Spectre' tiles including a 'Mystic' (dark grey lines) with display of all the key-points making quadrilaterals (7 red small and a green big one).

The level-1 cluster made of 9 'Spectre' tiles including a 'Mystic' (dark grey lines) with display of all the key-points making quadrilaterals (8 red small and a green big one).

The level-1 cluster made of 8 'Spectre' tiles including a 'Mystic' (red) with display of all the key-points making quadrilaterals (7 blue small and a magenta big one).

The level-1 cluster made of 9 'Spectre' tiles including a 'Mystic' (red) with display of all the key-points making quadrilaterals (8 blue small and a magenta big one).

The level-1 cluster made of 8 'Spectre' tiles including a 'Mystic' (red and dark grey lines) with display of all the key-points making quadrilaterals (7 blue small and a magenta big one).

The level-1 cluster made of 9 'Spectre' tiles including a 'Mystic' (red and dark grey lines) with display of all the key-points making quadrilaterals (8 blue small and a magenta big one).

The level 2 cluster made of 7 'Spectre' level 1 clusters with display of all the key-points making quadrilaterals (7 small and a big one).

The level 2 cluster made of 8 'Spectre' level 1 clusters with display of all the key-points making quadrilaterals (8 small and a big one).

The level 3 cluster made of 7 'Spectre' level 2 clusters with display of all the key-points making quadrilaterals (7 small and a big one).

The level 3 cluster made of 8 'Spectre' level 2 clusters with display of all the key-points making quadrilaterals (8 small and a big one).

A close-up of the 'EinStein' aperiodic 'Spectre' tiling.

Tridimensional display of the 'EinStein' aperiodic 'Spectre' tiling.

An aperiodic Penrose tiling of the plane.

An aperiodic non linear Penrose tiling of the plane.

An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose-.	An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose-.	An aperiodic Penrose tiling of the plane.
An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.	An aperiodic Penrose tiling of the plane.

An aperiodic non linear Penrose tiling of the plane.

The Penrose city.

A tridimensionally distorded aperiodic Penrose tiling of the plane.

An aperiodic Penrose tiling of the Golden Decagon -a Tribute to Piet Mondrian and Roger Penrose-.

An hexagonal tiling of the hyperbolic Poincaré disk -iteration 5- -a Tribute to Piet Mondrian and Henri Poincaré-.

Meshings [Maillages]:

Voronoi diagrams.

Heterogeneous meshing of a square.

Homogeneous/heterogeneous meshing of a square.

Untitled 0616.

Homogeneous meshing of a cube.

Causal set obtained by means of an homogeneous random meshing of a cube.	Causal set obtained by means of an homogeneous random meshing of a cube.	Homogeneous random meshing of a cube.	Heterogeneous -tridimensional fractal field- random meshing of a cube.	Heterogeneous -tridimensional fractal field- random meshing of a cube.
Heterogeneous -tridimensional fractal field- random meshing of a cube.	Heterogeneous -tridimensional gaussian field- random meshing of a cube.	Heterogeneous -tridimensional gaussian field- random meshing of a cube.	Heterogeneous -tridimensional gaussian field- random meshing of a cube.	Heterogeneous -tridimensional anti-gaussian field- random meshing of a cube.
Heterogeneous -tridimensional anti-gaussian field- random meshing of a cube.	Heterogeneous -tridimensional anti-gaussian field- random meshing of a cube.

Complexity, Connectivity and Consciousness -64 nodes-.	'Organic' network -64 nodes-.	'Organic' network -64 nodes-.	'Organic' network -64 nodes-.	Complexity, Connectivity and Consciousness -512 nodes-.
Complexity, Connectivity and Consciousness -4096 nodes-.

Homogeneous random non linear meshing of a cube.	Homogeneous random non linear meshing of a cube.	Heterogeneous -tridimensional fractal field- random non linear meshing of a cube.	Heterogeneous -tridimensional fractal field- random non linear meshing of a cube.	Heterogeneous -tridimensional gaussian field- random non linear meshing of a cube.
Heterogeneous -tridimensional gaussian field- random non linear meshing of a cube.	Heterogeneous -tridimensional anti-gaussian field- random non linear meshing of a cube.	Heterogeneous -tridimensional anti-gaussian field- random non linear meshing of a cube.

Heterogeneous meshing of a fractal surface.

Deterministic Trees [Arbres déterministes]:

A perfect bidimensional fractal tree and the self-similarity.

A binary tree.

A binary tree with 256 leaves.

Artistic view of a binary tree with 256 leaves.

A binary tree with 4096 leaves.

A binary tree with 256 leaves.

A vibrating ternary tree.

The golden binary tree.

Non Deterministic Trees [Arbres non déterministes]:

A random bidimensional fractal tree and the self-similarity.

A random tridimensional fractal tree and the self-similarity.

A random tridimensional binary tree.

Quadrangulation [Quadrangulation]:

The Schaeffer bijection.

Artistic view of the Schaeffer bijection.

Periodical quadrangulation of a square -18x18-.

Random quadrangulation of a square -18x18-.

Periodical quadrangulation of a cube -8x8x8-.

Random quadrangulation of a cube -8x8x8-.

Periodical quadrangulation of a cube -18x18x18-.

Random quadrangulation of a cube -18x18x18-.

Regular quadrangulation of the surface of a cylinder -18x18-.

Random quadrangulation of the surface of a cylinder -18x18-.

Regular quadrangulation of the volume of a cylinder -8x8x8-.

Random quadrangulation of the volume of a cylinder -8x8x8-.

Regular quadrangulation of the volume of a cylinder -18x18x18-.

Random quadrangulation of the volume of a cylinder -18x18x18-.

Regular quadrangulation of the surface of a sphere -18x18-.

Random quadrangulation of the surface of a sphere -18x18-.

Regular quadrangulation of the volume of a sphere -8x8x8-.

Random quadrangulation of the volume of a sphere -8x8x8-.

Regular quadrangulation of the volume of a sphere -18x18x18-.

Random quadrangulation of the volume of a sphere -18x18x18-.

'Regular' quadrangulation of the volume of a 'crumpled' sphere -18x18x8-.

Random quadrangulation of the volume of a 'crumpled' sphere -18x18x8-.

Regular quadrangulation of the surface of a torus -18x18-.

Random quadrangulation of the surface of a torus -18x18-.

Regular quadrangulation of the volume of a torus -8x8x8-.

Random quadrangulation of the volume of a torus -8x8x8-.

Regular quadrangulation of the volume of a torus -18x18x18-.

Random quadrangulation of the volume of a torus -18x18x18-.

'Regular' quadrangulation of the surface of a 'crumpled' torus -18x18-.

Random quadrangulation of the surface of a 'crumpled' torus -18x18-.

Triangulation [Triangulation]:

Simple random triangulation of a square -18x18-.

Double random triangulation of a square -18x18-.

Simple random triangulation of the surface of a sphere -18x18-.

Double random triangulation of the surface of a sphere -18x18-.

Simple random triangulation of the volume of a 'crumpled' sphere -18x18x8-.

Double random triangulation of the volume of a 'crumpled' sphere -18x18x8-.

Simple random triangulation of the surface of a cylinder -18x18-.

Double random triangulation of the surface of a cylinder -18x18-.

Simple random triangulation of the surface of a torus -18x18-.

Double random triangulation of the surface of a torus -18x18-.

Simple random triangulation of the surface of a 'crumpled' torus -18x18-.

Double random triangulation of the surface of a 'crumpled' torus -18x18-.

Random Surfaces [Surfaces Aléatoires]:

Tridimensional fractal surface (bird's-eye view).

Tridimensional 'smooth' fractal surface.

Two tridimensional 'smooth' fractal surfaces.

Tridimensional 'smooth' fractal surface with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-.

'Fractal set of ropes' based on a plane.

Fractal 'celestial body' based on a sphere.

Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Fractal 'celestial body' based on a sphere.

Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Fractal 'celestial body' based on a sphere.

Fractal 'celestial body' based on a sphere with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

'Fractal set of ropes' based on a plane.

'Fractal set of ropes' based on a plane with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Fractal 'celestial body' based on a torus.

Fractal 'celestial body' based on a torus with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Fractal 'celestial body' based on a torus.

Fractal 'celestial body' based on a torus and a tridimensional intertwining inside a sphere.	Fractal 'celestial body' based on a torus and a tridimensional intertwining inside a sphere.	Fractal 'celestial body' based on a torus and a tridimensional intertwining inside a sphere.	Fractal 'celestial body' based on a torus and a tridimensional intertwining inside a sphere.	Fractal 'celestial body' based on a torus and a 'recursive' -one iteration-' tridimensional intertwining inside a sphere.
Fractal 'celestial body' based on a torus and a 'recursive' -two iterations-' tridimensional intertwining inside a sphere.	Fractal 'celestial body' based on a torus and a tridimensional intertwining based on the Bonan-Jeener-Klein triple bottle inside a sphere.	Fractal 'celestial body' based on a torus and a tridimensional intertwining based on a sphere inside a sphere.

Fractal 'celestial body' based on a torus and a tridimensional intertwining inside a torus.

Fractal 'celestial body' based on a torus and a tridimensional intertwining inside the Bonan-Jeener-Klein triple bottle.

Fractal 'celestial body' based on the Bonan-Jeener-Klein triple bottle.

Fractal 'celestial body' based on the Bonan-Jeener-Klein triple bottle with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-.

Paradoxes [Paradoxes]:

The Simpson paradox.

The Bertrand paradox.

The extended Bertrand paradox.

Miscellaneous [Divers]:

Mathematics: an infinite pyramidal structure built on a few axioms.

A bijection.

To count the members of a set.

How to compute 'pi' with a gun.

A random permutation of pixel blocks of a bidimensional field.

A random permutation of pixel blocks of an aperiodic Penrose tiling of the plane.

Tridimensional display of two intricated random labyrinths -the wide one and the narrow one-.

Two intricated random labyrinths -the wide one and the narrow one-.

The 64 first lines of the Pascal's Triangle -modulo 2-.

The 64 first lines of the Pascal's Triangle -modulo 3-.

The 64 first lines of the Pascal's Triangle -modulo 5-.

The 64 first lines of the Pascal's Triangle -modulo 7-.

The 64 first lines of the Pascal's Triangle.

A 4-cube -an hypercube-.

A distorded -for the sake of display- 5-cube -an hyperhypercube-.

Table mountain.

Three hexagons and the twenty-eight first strictly positive integer numbers -nine of them being prime numbers-.

Artistic display of a Sudoku grid.

[Visual access to more Number Theory pictures [Accès visuel à plus d'images de Théorie des Nombres]]
[See the Number Theory Slide Show [Visionnez le Diaporama de Théorie des Nombres]]

Copyright © Jean-François COLONNA, 2000-2024.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2000-2024.

Jean-François COLONNA [Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France france telecom, France Telecom R&D

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

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

Rational Numbers [Nombres Rationnels]:

Real Numbers and Random Walks [Nombres Réels et Marches Aléatoires]:

Most Important Transcendent Numbers [Les nombres transcendants les plus importants]:

π [π]:

The Champernowne Number [Le nombre de Champernowne]:

Miscellaneous Numbers [Nombres Divers]:

Real Numbers, Spirals, Helix and more... [Nombres Réels, Spirales, Hélices et plus...]:

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

Extended Arithmetic [Arithmétique étendue]:

Fractal Sets [Ensembles Fractals]:

Space Filling Curves [Courbes remplissant l'Espace]:

Bidimensional Space Filling Curves [Courbes Bidimensionnelles remplissant l'Espace]:

Tridimensional Space Filling Curves [Courbes Tridimensionnelles remplissant l'Espace]:

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

Euclidian Geometry [Géométrie Euclidienne]:

Spherical Geometry [Géométrie Sphérique]:

Hyperbolic Geometry [Géométrie Hyperbolique]:

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

Gamma Function [La Fonction Gamma]:

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

Functions of an Octonionic Variable [Fonctions d'une Variable Octonionique]:

Prime Numbers [Les Nombres Premiers]:

The Continuum Hypothesis [L'Hypothèse du Continu]:

The Goldbach Conjecture [La Conjecture de Goldbach]:

The ABC Conjecture [La Conjecture ABC]:

The Syracuse Conjecture [La Conjecture de Syracuse]:

The Additive and Multiplicative Persistence Conjectures [Les Conjectures de Persistance Additive et Multiplicative]:

Elliptic Curves [Les Courbes Elliptiques]:

Distribution of N Points on a Sphere [Répartition de N Points sur une Sphère]:

Point Localization [Localisation de Points]:

Bi- and tridimensional Reflections [Réflexions bi- et tridimensionnelles]:

Knots [Nœuds]:

Monodimensional Binary Cellular Automata [Automates Cellulaires Binaires Monodimensionnels ]:

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

Bidimensional Cellular Automata -the bidimensional John Conway's life game and more- [Automates Cellulaires Bidimensionnels -le Jeu de la Vie Bidimensionnel de John Conway et plus-]:

Tridimensional Cellular Automata -the tridimensional John Conway's life game and more- [Automates Cellulaires Tridimensionnels -le Jeu de la Vie Tridimensionnel de John Conway et plus-]:

Voronoi Diagrams [Diagrammes de Voronoï]:

Conformal Transformations [Transformations Conformes]:

Polygons and more [Polygones et plus]:

Tilings [Pavages]:

Meshings [Maillages]:

Deterministic Trees [Arbres déterministes]:

Non Deterministic Trees [Arbres non déterministes]:

Quadrangulation [Quadrangulation]:

Triangulation [Triangulation]:

Random Surfaces [Surfaces Aléatoires]:

Paradoxes [Paradoxes]:

Miscellaneous [Divers]:

Copyright © Jean-François COLONNA, 2000-2024. Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2000-2024.

Copyright © Jean-François COLONNA, 2000-2024. Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2000-2024.

Jean-François COLONNA
[Contact me]

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
france telecom, France Telecom R&D

Copyright © Jean-François COLONNA, 2000-2024.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2000-2024.

Copyright © Jean-François COLONNA, 2000-2024.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2000-2024.