Gallery (306 different pictures):

Pi

[Galerie (306 images différentes) : Pi]

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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Pi [Pi]:

Self-Portraits [Auto-Portraits]	Artistic Views [Vues artistiques]	Base 2 [Base 2]	Base 3 [Base 3]	Base 4 -Random Walk- [Base 4 -Marche Aléatoire-]
Base 6 -Random Walk- [Base 6 -Marche Aléatoire-]	Base 10 -Random Walk- [Base 10 -Marche Aléatoire-]	Base 10 -Archimedes Spiral, color=F(value)- [Base 10 -Spirale d'Archimède, couleur=F(valeur)-]	Base 10 -Archimedes Spiral, color=F(rank)- [Base 10 -Spirale d'Archimède, couleur=F(rang)-]	Base 10 -Various Spirals, color=F(value)- [Base 10 -Spirales variées, couleur=F(valeur)-]
Base 10 -Helix- [Base 10 -Hélice-]	Fractal Squares [Carrés fractals]	Miscellaneous Fractal Structures [Diverses structures fractales]	Bidimensional Hilbert Curve [Courbes d'Hilbert bidimensionnelles]	Tridimensional Hilbert Curve [Courbes d'Hilbert tridimensionnelles]
Miscellaneous [Divers]	Best Of and Comments [Best Of et Commentaires] Chessboard [échiquier] -base 2- Random Walk [Marche Aléatoire] -base 6- Random Walk [Marche Aléatoire] -base 10- Archimedes Spiral [Spirale d'Archimède] -base 10- Helix [Hélice] -base 10- ADN Helix [Hélice ADN] -base 10- Menger Sponge [Eponge de Menger] -base 2- Bidimensional Hilbert Curve [Courbes d'Hilbert bidimensionnelles] -base 10- Impossible Structures [Structures Paradoxales] -base 10-	The Final Choice [Le choix final]

Pi [Pi]:

[More information]

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.
A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.
A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.

A set of 4x3 stereograms displaying a tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.

Base 10 -Archimedes Spiral, color=F(value)- [Base 10 -Spirale d'Archimède, couleur=F(valeur)-]:

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 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 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-.

Base 10 -Archimedes Spiral, color=F(rank)- [Base 10 -Spirale d'Archimède, couleur=F(rang)-]:

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 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 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-.

Base 10 -Various Spirals, color=F(value)- [Base 10 -Spirale variées, couleur=F(valeur)-]:

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-.

Base 10 -Helix- [Base 10 -Hélice-]:

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 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 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 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'-.

Fractal Squares [Carrés fractals]:

A Fractal Square -iteration 1-.

A Fractal Square -iteration 2-.

A Fractal Square -iteration 3-.

Tridimensional display of a Fractal Square -iteration 1 to 3-.

Miscellaneous Fractal Structures [Diverses structures fractales]:

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 54 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 81 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 108 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.
An extended Menger Sponge -iteration 6- displaying the 15.750.801 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 7- displaying the 211.210.335 first digits -base 2- of 'pi'.

A 4x4 Sierpinski Carpet -iteration 4- displaying the number 3.141.

A 5x5 Sierpinski Carpet -iteration 4- using the 9 first prime numbers and displaying the number 3.141.

A 3x3x3 Menger Sponge -iteration 4- using the 9 first prime numbers and displaying the number 3.141.

A 3x3x3 Menger Sponge -iteration 1- displaying the number 3.

A 3x3x3 Menger Sponge -iteration 2- displaying the number 3.1.

A 3x3x3 Menger Sponge -iteration 3- displaying the number 3.14.

A 3x3x3 Menger Sponge -iteration 4- displaying the number 3.141.

A 3x3x3 Menger Sponge -iteration 5- displaying the number 3.1415.

A 3x3x3 Menger Sponge -iteration 6- displaying the number 3.14159.

A double spherical cross-section inside a 3x3x3 Menger Sponge -iteration 4- displaying the number 3.141.

A 3x3x3 Menger Sponge -iteration 4- using the 9 first prime numbers and displaying the number 6.858 (=9.999-3.141).

A 3x3x3 Menger Sponge -iteration 6- displaying the number 6.85840 (=9.99999-3.14159).

Bidimensional Hilbert Curve [Courbes d'Hilbert bidimensionnelles]:

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 256 first digits -base 10- of 'pi' on a Bidimensional Hilbert Curve -iteration 4- mapped on a sphere.

Tridimensional Hilbert Curve [Courbes d'Hilbert bidimensionnelles]:

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-.

Miscellaneous [Divers]:

How to compute 'pi' with a gun.

The 120 first digits of 'pi' displayed as an Impossible Structure.

The 500 first digits of 'pi' displayed as an Impossible Structure.

The 120 first digits of 'pi/2' displayed as an Impossible Structure.

The 500 first digits of 'pi/2' displayed as an Impossible Structure.

The 120 first digits of 'pi/3' displayed as an Impossible Structure.

The 500 first digits of 'pi/3' displayed as an Impossible Structure.

The 120 first digits of 'pi/4' displayed as an Impossible Structure.

The 500 first digits of 'pi/4' displayed as an Impossible Structure.

The 120 first digits of 'pi/5' displayed as an Impossible Structure.

The 500 first digits of 'pi/5' displayed as an Impossible Structure.

The 120 first digits of 'pi/6' displayed as an Impossible Structure.

The 500 first digits of 'pi/6' displayed as an Impossible Structure.

The 120 first digits of 'pi' displayed as an Impossible Structure.

The 500 first digits of 'pi' displayed as an Impossible Structure.

The 180 first digits of 'pi' displayed as a Labyrinth.

The 81 first digits of 'pi' displayed as sets of concentric circles (bird's-eye view).

The 81 first digits of 'pi' displayed as sets of concentric circles.

The 1089 first digits of 'pi' displayed as sets of concentric circles (bird's-eye view).

The 1089 first digits of 'pi' displayed as sets of concentric circles.

The 1089 first digits of 'pi' displayed as sets of concentric circles (bird's-eye view).

The 1089 first digits of 'pi' displayed as sets of concentric circles.

The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).

The 81 first digits of 'pi' displayed as sets of concentric various curves.

The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).

The 81 first digits of 'pi' displayed as sets of concentric various curves.

The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).	The 81 first digits of 'pi' displayed as sets of concentric various curves.	The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).	The 81 first digits of 'pi' displayed as sets of concentric various curves.	The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).
The 81 first digits of 'pi' displayed as sets of concentric various curves.

The 1089 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).

The 1089 first digits of 'pi' displayed as sets of concentric various curves.

The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).

The 1089 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).

The 81 first digits of 'pi' displayed as sets of concentric various curves (bird's-eye view).

Best Of and Comments [Best Of et Commentaires]:

Un échiquier géant et chaotique:

Les 262.144 premièrs chiffres de π en base 2 (11,0010010000111111011010101...) sont marqués, en partant du haut à gauche, ligne après ligne, par des petits carrés noirs pour les "0" et blancs pour les "1".

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

Une marche aléatoire en base 6:

Une suite de chiffres en base 6 peut être utilisée pour définir la trajectoire d'un point P dans un espace tridimensionnel à partir d'une origine arbitraire. Pour ce faire, à chacun des six chiffres possibles {0,1,2,3,4,5} est associé un déplacement élémentaire {droite,gauche,haut,bas,devant,derrière}. Les 12.849 premières décimales de π en base 6 (05033005141512...) peuvent donc alors être utilisées pour définir pas à pas la trajectoire du point P suivant les conventions précédentes. La structure obtenue a toutes les apparences d'une marche aléatoire.

A tridimensional pseudo-random walk defined by means of 'pi': 050330... -12.849 digits, -base 6- into 050330... -12.849 digits, -base 6-.

Une marche aléatoire en base 10:

Les 90.000 premiers chiffres de π en base 10 (3.14159265358979...) peuvent être utilisées pour définir la trajectoire d'un point P dans un espace tridimensionnel à partir d'une origine arbitraire. Pour ce faire, ils sont regroupés par paquets de trois ([314] [159] [265] [358] [979]...), chaque paquet {C₁,C₂,C₃} définissant un déplacement élémentaire {D_x,D_y,D_z}={kC₁,kC₂,kC₃} de P, k désignant une constante arbitraire. La structure obtenue a toutes les apparences d'une marche aléatoire.

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.
A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.	A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.

A tridimensional pseudo-random walk defined by means of 'pi': 3.141592... -90.000 digits, -base 10- with 30.000 time steps.

Une spirale d'Archimède:

Les premiers chiffres de π (3,14159265358979...) sont représentés sur une spirale d'Archimède par des pics dont la couleur et la hauteur sont fonctions de chacun d'eux.

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 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-.

Une hélice:

Les premiers chiffres de π (3,14159265358979312...) sont marqués par des points sur une hélice qui sont joints les uns apres les autres.

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

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

The 20 first digits {141592...} of 'pi' displayed on an helix -orange-.

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

l'ADN des Mathématiques et de la Physique:

Une double hélice assez semblable à celle de l'ADN est construite et sur chacune des "bases" des deux brins sont portés respectivement les chiffres de π (3,14159265358979312...) et de 2π (6.283185307179586...). Cette image nous rappelle l'omniprésence de π en Mathématiques, mais aussi en Physique.

The DNA of Mathematics -the 480 first digits of 'pi' (3,141592...) and '2.pi' (6,283185...)-.

Une éponge de Menger d'apparence aléatoire:

L'éponge de Menger

classique est définie de la façon suivante: une cube K est découpé régulièrement en 3x3x3=27 cubes C. Le cube C au centre de K est supprimé, ainsi que les 6 qui sont au centre des 6 faces de K. Ainsi, on a supprimé 7 cubes C. Et cette procédure est répétée sur les 27-7=20 cubes C restant. Il est évidemment possibles de changer les régles et c'est ce qui est fait ici. A chaque fois que l'on devra décider de subdiviser un cube ou bien de le supprimer, un chiffre de π en base 2 (11,0010010000111111011010101...) sera prélevé et un '1' provoquera la subdivision, alors qu'un '0' induira la suppression du cube courant. La structure obtenue est très irrégulière et les chiffres peuvent être récupérés, mais cela est très difficile de par l'imbrication des cubes à toutes les échelles...

A la limite (au sens mathématique du terme) ce type de représentation permettrait de voir π, sans approximation, à l'intérieur d'un espace fini tridimensionnel.

An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'.
An extended Menger Sponge -iteration 6- displaying the 15.750.801 first digits -base 2- of 'pi'.	An extended Menger Sponge -iteration 7- displaying the 211.210.335 first digits -base 2- of 'pi'.

Les courbes remplissantes

: Lors de ses travaux sur les infinis, Georg Cantor a démontré que R, R², R³,... avaient le même cardinal. Cela a ouvert la porte aux courbes passant par tous les points d'un carré. Sur cette courbe de Hilbert

les 16.384 premières décimales de π (14159265358979312...) sont materialisées en plaçant en chacun de ses sommets une boule dont le rayon et la couleur sont fonctions de leur valeur.

A la limite (au sens mathématique du terme) ce type de représentation permettrait de voir π, sans approximation, à l'intérieur d'un espace fini bidimensionnel.

The 16.384 first digits -base 10- of 'pi' (141592...) on a Bidimensional Hilbert Curve -iteration 7-.

Des structures paradoxales basées sur π

: Une spirale "carrée" est tracée sur un quadrillage et en chacun de ses nœuds, en partant de l'origine, les premiers chiffres de π/n (π=3,14159265358979312... et n variant de 1 à 7 par la suite) sont marqués en étant de plus associés à des symboles graphiques choisis dans l'ensemble suivant:

0	1	2	3	4
5	6	7	8	9

L'assemblage bidimensionnel de ces symboles donne en général naissance à des structures d'apparences tridimensionnelles, mais impossibles à construire physiquement et qui sont dites paradoxales.

The 500 first digits of 'pi/1' displayed as an Impossible Structure.

The 500 first digits of 'pi/2' displayed as an Impossible Structure.

The 500 first digits of 'pi/3' displayed as an Impossible Structure.

The 500 first digits of 'pi/4' displayed as an Impossible Structure.

The 500 first digits of 'pi/5' displayed as an Impossible Structure.

The 500 first digits of 'pi/6' displayed as an Impossible Structure.

The 500 first digits of 'pi/7' displayed as an Impossible Structure.