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: obtenue à partir de la courbe génératrice .

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: obtenue à partir de la courbe génératrice .

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): .


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




The tridimensional Hilbert Curves:

Let's C1(T) being a parametric curve 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 in order to understand the geometrical meaning of the 8 transformations and of their order .


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

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


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









[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 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 in order to understand the geometrical meaning of the 4 transformations and of their order .


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

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


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





















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




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

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







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




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

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


(CMAP28 WWW site: this page was created on 05/16/2022 and last updated on 07/04/2023 12:14:01 -CEST-)



[See the generator of this picture [Voir le générateur de cette image]]

[See all related pictures (including this one) [Voir toutes les images associées (incluant celle-ci)]]

[Please visit the related DeterministicFractalGeometry picture gallery [Visitez la galerie d'images DeterministicFractalGeometry associée]]
[Please visit the related ImagesDesMathematiques picture gallery [Visitez la galerie d'images ImagesDesMathematiques associée]]
[Please visit the related NumberTheory picture gallery [Visitez la galerie d'images NumberTheory associée]]
[Please visit the related Tributes picture gallery [Visitez la galerie d'images Tributes associée]]

[Go back to AVirtualMachineForExploringSpaceTimeAndBeyond [Retour à AVirtualMachineForExploringSpaceTimeAndBeyond]]

[The Y2K Bug [Le bug de l'an 2000]]

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[Mail [Courrier]]
[About Pictures and Animations [A Propos des Images et des Animations]]


Copyright © Jean-François COLONNA, 2022-2023.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2022-2023.