Hilbert Curves and Infinite Knots

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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[en français/in french]

In the year 1890 Giuseppe Peano discovered curves that are going through all the points of a square (and of a cube) showing then continuous surjections between [0,1] and [0,1]x[0,1] (and [0,1]x[0,1]x[0,1]). During the following years, David Hilbert defined more such curves that can be generalized.

The bidimensional Hilbert Curves:

C₁(T)

a parametric curve A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} -iteration 1-

∈

X₁(T)

∈

Y₁(T)

∈

                    X₁(T=0)=0 Y₁(T=0)=0 (lower left corner)

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

C_i(T)

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

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

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

C₁(T) curve A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} -iteration 1-

4 transformations A Bidimensional Hilbert-like Curve defined with {X2(...),Y2(...)} -iteration 2-

order

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]

The construction of the bidimensional Hilbert Curve -iteration 3-

The construction of the bidimensional Hilbert Curve -iteration 4-

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-

Untitled 0598 -a Tribute to Robert & Sonia Delaunay-

==>
[iteration 11]

==>
[iteration 10]

==>
[iteration 9]

Untitled 0265 -a Tribute to Vassily Kandinsky-

==>
[iteration 10]

The tridimensional Hilbert Curves:

C₁(T)

a parametric curve A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} -iteration 1-

X₁(T)

Y₁(T)

Z₁(T)

                    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)

C_i(T)

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

                    if T ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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