AVirtualMachineForExploringSpaceTimeAndBeyond : Vibrating Tridimensional Hilbert Curve -iteration 4- (Courbe de Hilbert tridimensionnelle vibrante -iteration 4-).

Vibrating Tridimensional Hilbert Curve -iteration 4- [Courbe de Hilbert tridimensionnelle vibrante -itération 4-].

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

See the used color set to display the parameter T.

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 prime numbers.

See the used color set to display the parameter T.

See the used color set to display the pi digits.

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 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 prime numbers.

See the used color set to display the parameter T.

See the used color set to display the pi digits.

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 A_i 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 B_i E {0,1,2}

                    Y(T) = 0.C₁C₂C₃... E [0,1] with C_i E {0,1,2}

with:

                    B_n = A_2n-1 if A₂+A₄+...+A_2n-0 is even

                    B_n = 2-A_2n-1 otherwise

                    C_n = A_2n if A₁+A₃+...+A_2n-1 is even

                    C_n = 2-A_2n 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 A_i 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 B_i E {0,1,2}

                    Y(T) = 0.C₁C₂C₃... E [0,1] with C_i E {0,1,2}

                    Y(T) = 0.D₁D₂D₃... E [0,1] with D_i E {0,1,2}

with:

                    B_n = A_3n-2 if A₃+A₆+...+A_3n-0 is even

                    B_n = 2-A_3n-2 otherwise

                    C_n = A_3n-1 if A₂+A₅+...+A_3n-1 is even

                    C_n = 2-A_3n-1 otherwise

                    D_n = A_3n if A₁+A₄+...+A_3n-2 is even

                    D_n = 2-A_3n 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 C_i(T) (i >= 1) as follows :

                    C_i(T) = {X_i(T),Y_i(T)} E [0,1]x[0,1] --> C_i+1(T) = {X_i+1(T),Y_i+1(T)} E [0,1]x[0,1]

                    if T E [0,1/4[:
                              X_i+1(T) =   Y_i(4T-0)
                              Y_i+1(T) =   X_i(4T-0)
                                                   Transformation 1

                    if T E [1/4,2/4[:
                              X_i+1(T) =   X_i(4T-1)
                              Y_i+1(T) = 1+Y_i(4T-1)
                                                   Transformation 2

                    if T E [2/4,3/4[:
                              X_i+1(T) = 1+X_i(4T-2)
                              Y_i+1(T) = 1+Y_i(4T-2)
                                                   Transformation 3

                    if T E [3/4,1]:
                              X_i+1(T) = 2-Y_i(4T-3)
                              Y_i+1(T) = 1-X_i(4T-3)
                                                   Transformation 4

Please note that 4=2^d 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 C_i(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 C_i(T) (i >= 1) as follows :

                    C_i(T) = {X_i(T),Y_i(T),Z_i(T)} E [0,1]x[0,1]x[0,1] --> C_i+1(T) = {X_i+1(T),Y_i+1(T),Z_i+1(T)} E [0,1]x[0,1]x[0,1]

                    if T E [0,1/8[:
                              X_i+1(T) =   X_i(8T-0)
                              Y_i+1(T) =   Z_i(8T-0)
                              Z_i+1(T) =   Y_i(8T-0)
                                                   Transformation 1

                    if T E [1/8,2/8[:
                              X_i+1(T) =   Z_i(8T-1)
                              Y_i+1(T) = 1+Y_i(8T-1)
                              Z_i+1(T) =   X_i(8T-1)
                                                   Transformation 2

                    if T E [2/8,3/8[:
                              X_i+1(T) = 1+X_i(8T-2)
                              Y_i+1(T) = 1+Y_i(8T-2)
                              Z_i+1(T) =   Z_i(8T-2)
                                                   Transformation 3

                    if T E [3/8,4/8[:
                              X_i+1(T) = 1+Z_i(8T-3)
                              Y_i+1(T) = 1-X_i(8T-3)
                              Z_i+1(T) = 1-Y_i(8T-3)
                                                   Transformation 4

                    if T E [4/8,5/8[:
                              X_i+1(T) = 2-Z_i(8T-4)
                              Y_i+1(T) = 1-X_i(8T-4)
                              Z_i+1(T) = 1+Y_i(8T-4)
                                                   Transformation 5

                    if T E [5/8,6/8[:
                              X_i+1(T) = 1+X_i(8T-5)
                              Y_i+1(T) = 1+Y_i(8T-5)
                              Z_i+1(T) = 1+Z_i(8T-5)
                                                   Transformation 6

                    if T E [6/8,7/8[:
                              X_i+1(T) = 1-Z_i(8T-6)
                              Y_i+1(T) = 1+Y_i(8T-6)
                              Z_i+1(T) = 2-X_i(8T-6)
                                                   Transformation 7

                    if T E [7/8,1]:
                              X_i+1(T) =   X_i(8T-7)
                              Y_i+1(T) = 1-Z_i(8T-7)
                              Z_i+1(T) = 2-Y_i(8T-7)
                                                   Transformation 8

Please note that 8=2^d 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 C_i(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 04/18/2012 and last updated on 11/08/2022 13:00:26 -CET-)

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The bidimensional Peano Surjection:

The tridimensional Peano Surjection:

The bidimensional Peano-like Curves:

The tridimensional Peano-like Curves:

Copyright © Jean-François Colonna, 2012-2022. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2012-2022.

Copyright © Jean-François Colonna, 2012-2022.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2012-2022.