Tridimensional Hilbert Curve defined by means of the [0,1] --> [0,1]x[0,1]x[0,1] Peano Surjection -T defined with 6 digits- [Courbe de Hilbert tridimensionnelle définie grâce à la surjection [0,1] --> [0,1]x[0,1]x[0,1] de Peano -T défini avec 6 décimales-].

### 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 Ai 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 Bi E {0,1,2}
```
```                    Y(T) = 0.C1C2C3... E [0,1] with Ci E {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 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 Ai 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 Bi E {0,1,2}
```
```                    Y(T) = 0.C1C2C3... E [0,1] with Ci E {0,1,2}
```
```                    Y(T) = 0.D1D2D3... E [0,1] with Di E {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 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.

See various tridimensional 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 the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.

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

(CMAP28 WWW site: this page was created on 04/08/2022 and last updated on 06/01/2022 09:07:24 -CEST-)

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