The Tridimensional [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 3 digits- [La surjection tridimensionnelle [0,1] --> [0,1]x[0,1]x[0,1] de Peano -T défini avec 3 décimales-].

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}:

[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}: