A Tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} and based on an 'open' 3-foil torus knot -iteration 3- [Une courbe tridimensionnelle du type Hilbert définie avec {X3(...),Y3(...),Z3(...)} et basé sur un nœud '3-trèfle' torique 'ouvert' -itération 3- ]

A Tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} and based on an 'open' 3-foil torus knot -iteration 3- [Une courbe tridimensionnelle du type Hilbert définie avec {X3(...),Y3(...),Z3(...)} et basé sur un nœud '3-trèfle' torique 'ouvert' -itération 3-].




A la fin du dix-neuvième siècle, après que Georg Cantor ait montré que [0,1] et [0,1]x[0,1] avait même cardinal, Giuseppe Peano puis David Hilbert ont proposé des courbes continues passant par tous les points du carré unité. Voici une grossière approximation de l'une d'elle: Bidimensional Hilbert Curve -iteration 4- obtenue à partir de la courbe génératrice Bidimensional Hilbert Curve -iteration 1- .

Cela se generalise sans problèmes dans les dimensions supérieures et en particulier [0,1] et [0,1]x[0,1]x[0,1] ont le même cardinal. Ainsi, il est possible de disposer de courbes continues passant par tous les points du cube unité. Voici, là aussi, une grossière approximation de l'une d'elle: Tridimensional Hilbert Curve -iteration 3- obtenue à partir de la courbe génératrice Tridimensional Hilbert Curve -iteration 1- .

Mais il est évidemment possible d'utiliser des courbes génératrices différentes, mais à condition qu'elles respectent une contrainte essentielles concernant leurs deux extrémités: celles-ci doivent être localisée sur deux sommets voisins du cube. On pourra, par exemple, comme c'est le cas ici, utiliser un nœud de trèfle "ouvert" (c'est-à-dire que l'on a coupé): A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} and based on an 'open' 3-foil torus knot -iteration 1- .


[Plus d'informations sur ce sujet (en français/in french)]
[Plus d'informations sur ce sujet (en français/in french)]



See its construction:

The construction of a tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} and based on an 'open' 3-foil torus knot -iteration 3-



The tridimensional Hilbert Curves:

Let's C1(T) being a parametric curve A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} and based on an 'open' 3-foil torus knot -iteration 1- defined by means of 3 real functions of T (T ∈ [0,1]) X1(T) ∈ [0,1], Y1(T) ∈ [0,1] and Z1(T) ∈ [0,1] such as : 
                    X1(T=0)=0 Y1(T=0)=0 Z1(T=0)=0 (lower left foreground corner)

                    X1(T=1)=1 Y1(T=1)=0 Z1(T=1)=0 (lower right foreground corner)


Then one defines a sequence of curves Ci(T) (i >= 1) as follows : 
                    Ci(T) = {Xi(T),Yi(T),Zi(T)} ∈ [0,1]x[0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T),Zi+1(T)} ∈ [0,1]x[0,1]x[0,1]


                    if T ∈ [0,1/8[:
                              Xi+1(T) =   Xi(8T-0)
                              Yi+1(T) =   Zi(8T-0)
                              Zi+1(T) =   Yi(8T-0)
                                                   Transformation 1

                    if T ∈ [1/8,2/8[:
                              Xi+1(T) =   Zi(8T-1)
                              Yi+1(T) = 1+Yi(8T-1)
                              Zi+1(T) =   Xi(8T-1)
                                                   Transformation 2

                    if T ∈ [2/8,3/8[:
                              Xi+1(T) = 1+Xi(8T-2)
                              Yi+1(T) = 1+Yi(8T-2)
                              Zi+1(T) =   Zi(8T-2)
                                                   Transformation 3

                    if T ∈ [3/8,4/8[:
                              Xi+1(T) = 1+Zi(8T-3)
                              Yi+1(T) = 1-Xi(8T-3)
                              Zi+1(T) = 1-Yi(8T-3)
                                                   Transformation 4

                    if T ∈ [4/8,5/8[:
                              Xi+1(T) = 2-Zi(8T-4)
                              Yi+1(T) = 1-Xi(8T-4)
                              Zi+1(T) = 1+Yi(8T-4)
                                                   Transformation 5

                    if T ∈ [5/8,6/8[:
                              Xi+1(T) = 1+Xi(8T-5)
                              Yi+1(T) = 1+Yi(8T-5)
                              Zi+1(T) = 1+Zi(8T-5)
                                                   Transformation 6

                    if T ∈ [6/8,7/8[:
                              Xi+1(T) = 1-Zi(8T-6)
                              Yi+1(T) = 1+Yi(8T-6)
                              Zi+1(T) = 2-Xi(8T-6)
                                                   Transformation 7

                    if T ∈ [7/8,1]:
                              Xi+1(T) =   Xi(8T-7)
                              Yi+1(T) = 1-Zi(8T-7)
                              Zi+1(T) = 2-Yi(8T-7)
                                                   Transformation 8

Please note that 8=2d where d=3 is the space dimension.



See a special C1(T) curve A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} -iteration 1- in order to understand the geometrical meaning of the 8 transformations A Tridimensional Hilbert-like Curve defined with {X2(...),Y2(...),Z2(...)} -iteration 2- and of their order Tridimensional Hilbert Curve -iteration 1- .


Here are the four first tridimensional Hilbert curves with an increasing number of iterations :

Tridimensional Hilbert Curve -iteration 1- Tridimensional Hilbert Curve -iteration 2- Tridimensional Hilbert Curve -iteration 3- Tridimensional Hilbert Curve -iteration 4- [See the used color set to display the parameter T]

See the construction of one of them :

Empty Empty The construction of the tridimensional Hilbert Curve -iteration 3- Empty


Here are some examples of Hilbert-like tridimensional curves using different generating curves and in particular "open" knots :

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(...)} -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(...)} -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-
Empty Empty The construction of a tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} and based on an 'open' 3-foil torus knot -iteration 3- Empty
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-
Empty Empty The construction of a tridimensional Hilbert-like Curve defined with {X3(...),Y3(...),Z3(...)} and based on an 'open' 3-foil torus knot -iteration 3- Empty
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 {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-

[More information about Peano Curves and Infinite Knots -in english/en anglais-]
[Plus d'informations à propos des Courbes de Peano et des Nœuds Infinis -en français/in french-]











The bidimensional Hilbert Curves:

Let's C1(T) being a parametric curve A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} -iteration 1- defined by means of 2 real functions of T (T ∈ [0,1]) X1(T) ∈ [0,1] and Y1(T) ∈ [0,1] such as : 
                    X1(T=0)=0 Y1(T=0)=0 (lower left corner)

                    X1(T=1)=1 Y1(T=1)=0 (lower right corner)


Then one defines a sequence of curves Ci(T) (i >= 1) as follows : 
                    Ci(T) = {Xi(T),Yi(T)} ∈ [0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T)} ∈ [0,1]x[0,1]


                    if T ∈ [0,1/4[:
                              Xi+1(T) =   Yi(4T-0)
                              Yi+1(T) =   Xi(4T-0)
                                                   Transformation 1

                    if T ∈ [1/4,2/4[:
                              Xi+1(T) =   Xi(4T-1)
                              Yi+1(T) = 1+Yi(4T-1)
                                                   Transformation 2

                    if T ∈ [2/4,3/4[:
                              Xi+1(T) = 1+Xi(4T-2)
                              Yi+1(T) = 1+Yi(4T-2)
                                                   Transformation 3

                    if T ∈ [3/4,1]:
                              Xi+1(T) = 2-Yi(4T-3)
                              Yi+1(T) = 1-Xi(4T-3)
                                                   Transformation 4

Please note that 4=2d where d=2 is the space dimension.



See a special C1(T) curve A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} -iteration 1- in order to understand the geometrical meaning of the 4 transformations A Bidimensional Hilbert-like Curve defined with {X2(...),Y2(...)} -iteration 2- and of their order Bidimensional Hilbert Curve -iteration 1- .


Here are the five first bidimensional Hilbert curves with an increasing number of iterations :

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- [See the used color set to display the parameter T]

See the construction of some of them :

Empty Empty The construction of the bidimensional Hilbert Curve -iteration 3- The construction of the bidimensional Hilbert Curve -iteration 4- Empty


Here are some examples of Hilbert-like bidimensional curves using different generating curves :

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 {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 {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 von Koch Curve -iteration 1- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 2- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 3- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 4- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to the von Koch Curve -iteration 5-
A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 1- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 2- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 3- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 4- A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical impossible structure -iteration 5-
A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 1- A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 2- A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 3- A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 4- A Bidimensional non continous Hilbert-like Curve defined with {X1(...),Y1(...)} related to a periodical 'labyrinthic' structure -iteration 5-


Here is the "mapping" of a few pictures by means of a bidimensional Hilbert curve: :

width="TaBlE_WiDtH%"
Untitled 0598 -a Tribute to Robert & Sonia Delaunay- ==>
[iteration 11]		 Untitled 0639
width="TaBlE_WiDtH%"
Untitled 0198 ==>
[iteration 10]		 Untitled 0638
width="TaBlE_WiDtH%"
Rings ==>
[iteration 9]		 Untitled 0637
width="TaBlE_WiDtH%"
Untitled 0265 -a Tribute to Vassily Kandinsky- ==>
[iteration 10]		 Untitled 0636















See various Bidimensional Hilbert and Peano Curves (possibly including this one):

Bidimensional Hilbert Curve -iterations 1 to 5- Bidimensional Hilbert Curve -iterations 1 to 5-


See Bidimensional Hilbert Curves, their nodes being "loaded" with some data (related to prime numbers, real number digits,...):

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 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- See the used color set to display the parameter T.  
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- See the used color set to display the pi digits.  
Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 1- with display of Julia sets Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 2- with display of Julia sets Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 3- with display of Julia sets Journey on the Complex Plane by means of a Bidimensional Hilbert Curve -iteration 4- with display of Julia sets Empty






See various Tridimensional Hilbert and Peano Curves (possibly including this one):

Tridimensional Hilbert Curve -iterations 1 to 3- Tridimensional Hilbert Curve -iterations 1 to 3-


See Tridimensional Hilbert Curves, their nodes being "loaded" with some data (related to prime numbers, real number digits,...):

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- Empty  
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- Empty See the used color set to display the parameter T.  
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- Empty See the used color set to display the pi digits.


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