A tridimensional Hilbert-like curve defined with {X4(...),Y4(...),Z4(...)} and based on an 'open' 5-foil torus knot -iteration 4- [Une courbe tridimensionnelle du type Hilbert définie avec {X4(...),Y4(...),Z4(...)} et basé sur un nœud '5-trèfle' torique 'ouvert' -itération 4-].





The tridimensional Hilbert Curves:

Let's C1(T) being a parametric curve defined by means of 3 real functions of T (T E [0,1]) X1(T) E [0,1], Y1(T) E [0,1] and Z1(T) E [0,1 such as :
                    X1(T=0)=0 Y1(T=0)=0 Z1(T=0)=0 (lower left foreground corner)
                    X1(T=1)=0 Y1(T=1)=0 Z1(T=1)=1 (lower right foreground corner)


Then one defines a sequence of curves Ci(T) (i >= 1) as follows :
                    Ci(T) = {Xi(T),Yi(T),Zi(T)} E [0,1]x[0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T),Zi+1(T)} E [0,1]x[0,1]x[0,1]

                    if T E [0,1/8[:
                              Xi+1(T) =   Xi(8T-0)
                              Yi+1(T) =   Zi(8T-0)
                              Zi+1(T) =   Yi(8T-0)
                                                   Transformation 1
                    if T E [1/8,2/8[:
                              Xi+1(T) =   Zi(8T-1)
                              Yi+1(T) = 1+Yi(8T-1)
                              Zi+1(T) =   Xi(8T-1)
                                                   Transformation 2
                    if T E [2/8,3/8[:
                              Xi+1(T) = 1+Xi(8T-2)
                              Yi+1(T) = 1+Yi(8T-2)
                              Zi+1(T) =   Zi(8T-2)
                                                   Transformation 3
                    if T E [3/8,4/8[:
                              Xi+1(T) = 1+Zi(8T-3)
                              Yi+1(T) = 1-Xi(8T-3)
                              Zi+1(T) = 1-Yi(8T-3)
                                                   Transformation 4
                    if T E [4/8,5/8[:
                              Xi+1(T) = 2-Zi(8T-4)
                              Yi+1(T) = 1-Xi(8T-4)
                              Zi+1(T) = 1+Yi(8T-4)
                                                   Transformation 5
                    if T E [5/8,6/8[:
                              Xi+1(T) = 1+Xi(8T-5)
                              Yi+1(T) = 1+Yi(8T-5)
                              Zi+1(T) = 1+Zi(8T-5)
                                                   Transformation 6
                    if T E [6/8,7/8[:
                              Xi+1(T) = 1-Zi(8T-6)
                              Yi+1(T) = 1+Yi(8T-6)
                              Zi+1(T) = 2-Xi(8T-6)
                                                   Transformation 7
                    if T E [7/8,1]:
                              Xi+1(T) =   Xi(8T-7)
                              Yi+1(T) = 1-Zi(8T-7)
                              Zi+1(T) = 2-Yi(8T-7)
                                                   Transformation 8

Please note that 8=2d where d=3 is the space dimension.



See a special C1(T) curve in order to understand the geometrical meaning of the 8 transformations and of their order .


Here are the four first tridimensional Hilbert curves with an increasing number of iterations {1,2,3,4}:

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


Here are some examples of Hilbert tridimensional curves :







[More information about Peano Curves and Infinite Knots]


See the 5 first Ci(T) curves (i E {1,2,3,4,5}) the first one being an "open" 5-foil torus knot:

See the used color set to display the parameter T.








The bidimensional Hilbert Curves:

Let's C1(T) being a parametric curve defined by means of 2 real functions of T (T E [0,1]) X1(T) E [0,1] and Y1(T) E [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 Ci(T) (i >= 1) as follows :
                    Ci(T) = {Xi(T),Yi(T)} E [0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T)} E [0,1]x[0,1]

                    if T E [0,1/4[:
                              Xi+1(T) =   Yi(4T-0)
                              Yi+1(T) =   Xi(4T-0)
                                                   Transformation 1
                    if T E [1/4,2/4[:
                              Xi+1(T) =   Xi(4T-1)
                              Yi+1(T) = 1+Yi(4T-1)
                                                   Transformation 2
                    if T E [2/4,3/4[:
                              Xi+1(T) = 1+Xi(4T-2)
                              Yi+1(T) = 1+Yi(4T-2)
                                                   Transformation 3
                    if T E [3/4,1]:
                              Xi+1(T) = 2-Yi(4T-3)
                              Yi+1(T) = 1-Xi(4T-3)
                                                   Transformation 4

Please note that 4=2d where d=2 is the space dimension.



See a special C1(T) curve in order to understand the geometrical meaning of the 4 transformations and of their order .


Here are the five first bidimensional Hilbert curves with an increasing number of iterations {1,2,3,4,5}:

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


Here are some examples of Hilbert bidimensional curves :





See the 6 first Ci(T) curves (i E {1,2,3,4,5,6}):

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 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 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 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.


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