Tridimensional Hilbert Curve -iterations 1 to 3- [Courbe de Hilbert tridimensionnelle -itérations 1 à 3-].




See various tridimensional Hilbert and Peano curves (possibly including this one):



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

 
 
 



See various bidimensional Hilbert and Peano curves (possibly including this one):



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








The bidimensional Peano Surjection:

Peano defined the following surjection:
                    [0,1] --> [0,1]x[0,1]
Let's T being a real number defined using the base 3:
                    T    = 0.A1A2A3... E [0,1] with Ai E {0,1,2}
Let's X(T) and Y(T) being 2 real functions of T defined as:
                    X(T) = 0.B1B2B3... E [0,1] with Bi E {0,1,2}
                    Y(T) = 0.C1C2C3... E [0,1] with Ci E {0,1,2}
with:
                    Bn = A2n-1 if A2+A4+...+A2n-0 is even
                    Bn = 2-A2n-1 otherwise

                    Cn = A2n if A1+A3+...+A2n-1 is even
                    Cn = 2-A2n otherwise


These 2 functions X(T) and Y(T) are the coordinates of a point P(T) inside the [0,1]x[0,1] square. The displayed "curve" is the trajectory of P(T) -displayed as little spheres- when T varies from 0 (lower left corner) to 1-epsilon (upper right corner).


See some related pictures (possibly including this one):

See the used color set to display the parameter T.



The tridimensional Peano Surjection:

A tridimensional surjection can be defined:
                    [0,1] --> [0,1]x[0,1]x[0,1]
as a generalization of the bidimensional one.

Let's T being a real number defined using the base 3:
                    T    = 0.A1A2A3... E [0,1] with Ai E {0,1,2}
Let's X(T), Y(T) and Z(T) being 3 real functions of T defined as:
                    X(T) = 0.B1B2B3... E [0,1] with Bi E {0,1,2}
                    Y(T) = 0.C1C2C3... E [0,1] with Ci E {0,1,2}
                    Y(T) = 0.D1D2D3... E [0,1] with Di E {0,1,2}
with:
                    Bn = A3n-2 if A3+A6+...+A3n-0 is even
                    Bn = 2-A3n-2 otherwise

                    Cn = A3n-1 if A2+A5+...+A3n-1 is even
                    Cn = 2-A3n-1 otherwise

                    Dn = A3n if A1+A4+...+A3n-2 is even
                    Dn = 2-A3n otherwise


These 3 functions X(T), Y(T) and Z(T) are the coordinates of a point P(T) inside the [0,1]x[0,1]x[0,1] cube. The displayed "curve" is the trajectory of P(T) -displayed as little spheres- when T varies from 0 (lower left corner) to 1-epsilon (upper right corner).


See some related pictures (possibly including this one):

See the used color set to display the parameter T.








The bidimensional Peano-like Curves:

Let's C1(T) being a parametric curve defined by means of 2 real functions of T (T E [0,1]) X1(T) E [0,1] and Y1(T) E [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)} E [0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T)} E [0,1]x[0,1]

                    if T E [0,1/4[:
                              Xi+1(T) =   Yi(4T-0)
                              Yi+1(T) =   Xi(4T-0)
                                                   Transformation 1
                    if T E [1/4,2/4[:
                              Xi+1(T) =   Xi(4T-1)
                              Yi+1(T) = 1+Yi(4T-1)
                                                   Transformation 2
                    if T E [2/4,3/4[:
                              Xi+1(T) = 1+Xi(4T-2)
                              Yi+1(T) = 1+Yi(4T-2)
                                                   Transformation 3
                    if T E [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 .


See the 6 first Ci(T) curves (i E {1,2,3,4,5,6}):

See the used color set to display the parameter T.








The tridimensional Peano-like Curves:

Let's C1(T) being a parametric curve defined by means of 3 real functions of T (T E [0,1]) X1(T) E [0,1], Y1(T) E [0,1] and Z1(T) E [0,1 such as :
                    X1(T=0)=0 Y1(T=0)=0 Z1(T=0)=0 (lower left foreground corner)
                    X1(T=1)=0 Y1(T=1)=0 Z1(T=1)=1 (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)} E [0,1]x[0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T),Zi+1(T)} E [0,1]x[0,1]x[0,1]

                    if T E [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 E [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 E [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 E [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 E [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 E [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 E [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 E [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 .


See the 5 first Ci(T) curves (i E {1,2,3,4,5}):

See the used color set to display the parameter T.


(CMAP28 WWW site: this page was created on 12/15/2015 and last updated on 06/01/2022 09:07:08 -CEST-)



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

[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]]

[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, 2015-2022.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2015-2022.