Hilbert and Peano Space filling Curves and Beyond:
From Squares and Cubes to Surfaces (bidimensional Manifolds) and tridimensional Manifolds

Jean-François COLONNA
[Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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Contents:

1-The bidimensional Peano Surjection:
2-The tridimensional Peano Surjection:
3-The bidimensional Hilbert Curves:
4-The tridimensional Hilbert Curves:
5-Tridimensional Surfaces (Bidimensional Manifolds):
6-Tridimensional Manifolds:
7-Beyond:

1-The bidimensional Peano Surjection:

Giuseppe 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... ∈ [0,1] with A_i ∈ {0,1,2}

Let's X(T) and Y(T) being two real functions of T defined as :

                    X(T) = 0.B₁B₂B₃... ∈ [0,1] with B_i ∈ {0,1,2}

                    Y(T) = 0.C₁C₂C₃... ∈ [0,1] with C_i ∈ {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 two 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" -as little spheres- is the trajectory of P(T) when T varies from 0 (lower left corner) to 1-epsilon (upper right corner).

Here are the four first bidimensional Peano curves with an increasing number of digits {2,4,6,8}:

The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 2 digits-

The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 4 digits-

The [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 6 digits-

The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 8 digits-

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

2-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... ∈ [0,1] with A_i ∈ {0,1,2}

Let's X(T), Y(T) and Z(T) being three real functions of T defined as:

                    X(T) = 0.B₁B₂B₃... ∈ [0,1] with B_i ∈ {0,1,2}

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

                    Y(T) = 0.D₁D₂D₃... ∈ [0,1] with D_i ∈ {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 three 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).

Here are the three first tridimensional Peano curves with an increasing number of digits {3,6,9}:

The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 3 digits-

The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 6 digits-

The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 9 digits-

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

3-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 C_i(T) (i >= 1) as follows :

                    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

Please note that 4=2^d 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 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 :

The construction of the bidimensional Hilbert Curve -iteration 3-

The construction of the bidimensional Hilbert Curve -iteration 4-

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

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: :

Untitled 0598 -a Tribute to Robert & Sonia Delaunay-

==>
[iteration 11]

==>
[iteration 10]

==>
[iteration 9]

Untitled 0265 -a Tribute to Vassily Kandinsky-

==>
[iteration 10]

4-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 C_i(T) (i >= 1) as follows :

                    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

Please note that 8=2^d 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 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 :

The construction of the tridimensional Hilbert Curve -iteration 3-

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 {X3(...),Y3(...),Z3(...)} -iteration 3-

A Tridimensional Hilbert-like Curve defined with {X4(...),Y4(...),Z4(...)} -iteration 4-

The construction of 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 {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' 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-]

5-Tridimensional Surfaces (Bidimensional Manifolds):

Many surfaces -bidimensional manifolds- in a tridimensional space can be defined using a set of three equations:

                    X = F_x(u,v)

                    Y = F_y(u,v)

                    Z = F_z(u,v)

with:

                    u ∈ [U_min,U_max]

                    v ∈ [V_min,V_max]

For example:

                    F_x(u,v) = R.sin(u).cos(v)

                    F_y(u,v) = R.sin(u).sin(v)

                    F_z(u,v) = R.cos(u)

with:

                     u ∈ [0,pi]

                     v ∈ [0,2.pi]

defines a sphere with R as the radius and the origin of the coordinates as the center.

[U_min,U_max]*[V_min,V_max] then defined a bidimensional rectangular domain D.

                       v ^
                         |
                    V    |...... ---------------------------
                     max |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                    V    |...... ---------------------------
                     min |      :                           :
                         |      :                           :
                         O------------------------------------------------->
                                U                           U              u
                                 min                         max

Let's define a curve that fills the [0,1]x[0,1] square :

                         ^
                         |
                       1 |---------------
                         | +++++   +++++ |
                         | +   +   +   + |
                         | +   +++++   + |
                         | +           + |
                         | +++++   +++++ |
                         |     +   +     |
                         | +++++   +++++ |
                       0 O------------------------>
                         0               1

Obviously one can define a mapping between the [0,1]x[0,1] square and the [U_min,U_max]*[V_min,V_max] domain :

                       v ^
                         |
                    V    |...... ---------------------------
                     max |      | ++++++++         ++++++++ |
                         |      | +      +         +      + |
                         |      | +      +         +      + |
                         |      | +      +++++++++++      + |
                         |      | +           C           + |
                         |      | ++++++++         ++++++++ |
                         |      |        +         +        |
                         |      |        +         +        |
                         |      | ++++++++         ++++++++ |
                    V    |...... ---------------------------
                     min |      :                           :
                         |      :                           :
                         O------------------------------------------------->
                                U                           U              u
                                 min                         max

Then it suffices to display only the points {F_x(u,v),F_y(u,v),F_z(u,v)} with {u,v} on the preceding curve C to fill the surface with C....

Here are some examples of this process :

Surface

==>

C Curve

==>

Surface filling Curve

	==>		==>
	==>		==>
	==>		==>

A 'crumpled' sphere defined by means of three bidimensional fields

==>

Bidimensional Hilbert Curve -iteration 7-

==>

A 'crumpled' sphere described by means of a Bidimensional Hilbert Curve -iteration 7-

	==>		==>
	==>		==>
	==>		==>

	==>		==>
	==>		==>
	==>		==>

	==>		==>
	==>		==>
	==>		==>
	==>		==>
	==>		==>
	==>		==>
	==>		==>
	==>		==>
	==>		==>

	==>		==>
	==>		==>
	==>		==>

$A 'fractal plane' with overhangings defined by means of three bidimensional fields$

==>

$A 'fractal plane' with overhangings described by means of a Bidimensional Hilbert Curve -iteration 7-$

$A 'fractal plane' with overhangings defined by means of three bidimensional fields$

==>

$A 'fractal plane' with overhangings described by means of a Bidimensional Hilbert Curve -iteration 7-$

$A 'fractal plane' with overhangings defined by means of three bidimensional fields$

==>

$A 'fractal plane' with overhangings described by means of a Bidimensional Hilbert-like Curve -iteration 6-$

Tridimensional representation of a quadridimensional Calabi-Yau manifold

==>

Tridimensional representation of a quadridimensional Calabi-Yau manifold described by means of 5x5 Bidimensional Hilbert Curves -iteration 5-

6-Tridimensional Manifolds:

Many tridimensional manifolds in a tridimensional space can be defined using a set of three equations:

                    X = F_x(u,v,w)

                    Y = F_y(u,v,w)

                    Z = F_z(u,v,w)

with:

                    u ∈ [U_min,U_max]

                    v ∈ [V_min,V_max]

                    w ∈ [W_min,W_max]

[U_min,U_max]*[V_min,V_max]*[W_min,W_max] then defined a tridimensional rectangular domain D.

It is obvious to generalize the preceding bidimensional process in the tridimensional space...

Here are some examples of this process :

Tridimensional Manifold

==>

C Curve

==>

Tridimensional Manifold filling Curve

==>

A Ball described by means of a Tridimensional Hilbert Curve -iteration 2-

==>

A Ball described by means of a Tridimensional Hilbert Curve -iteration 3-

==>

A Ball described by means of a Tridimensional Hilbert Curve -iteration 4-

==>

A Ball described by means of a Tridimensional Hilbert-like Curve -iteration 1-

==>

A Ball described by means of a Tridimensional Hilbert-like Curve -iteration 2-

==>

A Ball described by means of a Tridimensional Hilbert-like Curve -iteration 3-

==>

A Ball described by means of a Tridimensional Hilbert-like Curve -iteration 4-

==>

A Ball described by means of an hypercube -iteration 1-

==>

A Ball described by means of an hypercube -iteration 2-

==>

A Ball described by means of an hypercube -iteration 3-

==>

A Ball described by means of an hypercube -iteration 4-

==>

A Ball described by means of an 'open' 3-foil torus knot -iteration 2-

==>

A Ball described by means of an 'open' 3-foil torus knot -iteration 3-

==>

A Ball described by means of an 'open' 3-foil torus knot -iteration 4-

==>

A parallelepipedic Torus described by means of a Tridimensional Hilbert Curve -iteration 4-

==>

A parallelepipedic Torus described by means of an 'open' 3-foil torus knot -iteration 4-

==>

A parallelepipedic Torus described by means of an hypercube -iteration 4-

==>

A Jeener-Möbius Tridimensional manifold described by means of a Tridimensional Hilbert Curve -iteration 4-

==>

A Jeener-Möbius Tridimensional manifold described by means of an 'open' 3-foil torus knot -iteration 4-

7-Beyond:

Instead of using space-filling curves in order to fill the {u,v} and {u,v,w} bi- and tridimensional domains, obviously one can use any means available and, for example, the bi- and the tridimensional brownian motions respectively....

Here are some examples of these extended processes :

Bidimensional brownian motion on a sphere

Bidimensional brownian motion on a torus

Bidimensional brownian motion on the Möbius strip

Bidimensional brownian motion on the Klein bottle

Bidimensional brownian motion on the Bonan-Jeener-Klein triple bottle

Bidimensional brownian motion on the Boy surface

Bidimensional closed self-avoiding brownian motion on a sphere

Bidimensional closed self-avoiding brownian motion on a torus

Bidimensional closed self-avoiding brownian motion on the Möbius strip

Bidimensional closed pseudo-self-avoiding brownian motion on the Klein bottle

Bidimensional closed pseudo-self-avoiding brownian motion on the Bonan-Jeener-Klein triple bottle

Bidimensional closed pseudo-self-avoiding brownian motion on the Boy surface

Hilbert and Peano Space filling Curves and Beyond: From Squares and Cubes to Surfaces (bidimensional Manifolds) and tridimensional Manifolds

Jean-François COLONNA [Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

1-The bidimensional Peano Surjection:

2-The tridimensional Peano Surjection:

3-The bidimensional Hilbert Curves:

4-The tridimensional Hilbert Curves:

5-Tridimensional Surfaces (Bidimensional Manifolds):

6-Tridimensional Manifolds:

7-Beyond:

Copyright © Jean-François COLONNA, 2023-2026. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2023-2026.

Hilbert and Peano Space filling Curves and Beyond:
From Squares and Cubes to Surfaces (bidimensional Manifolds) and tridimensional Manifolds

Jean-François COLONNA
[Contact me]

Copyright © Jean-François COLONNA, 2023-2026.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2023-2026.