The Bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 2 digits- [La surjection bidimensionnelle [0,1] --> [0,1]x[0,1] de Peano -T défini avec 2 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 Ai ∈ {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 Bi ∈ {0,1,2}
```
```                    Y(T) = 0.C1C2C3... ∈ [0,1] with Ci ∈ {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 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 Ai ∈ {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 Bi ∈ {0,1,2}
```
```                    Y(T) = 0.C1C2C3... ∈ [0,1] with Ci ∈ {0,1,2}
```
```                    Y(T) = 0.D1D2D3... ∈ [0,1] with Di ∈ {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 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}:

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

See various Bidimensional Hilbert and Peano Curves (possibly including this one):

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

See various Tridimensional Hilbert and Peano Curves (possibly including this one):

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

(CMAP28 WWW site: this page was created on 03/24/2022 and last updated on 07/04/2023 12:13:26 -CEST-)

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