The bidimensional [0,1] --> [0,1]x[0,1] Peano Surjection -T defined with 4 digits- [La surjection bidimensionnelle [0,1] --> [0,1]x[0,1] de Peano -T défini avec 4 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 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.


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.

 
 
 



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