Bidimensional Hilbert Curve -iteration 4- [Courbe de Hilbert bidimensionnelle -itération 4-].

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.

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]

The bidimensional Hilbert Curves:

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

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

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

Here are some examples of Hilbert-like bidimensional curves using different generating curves :

The tridimensional Hilbert Curves:

Let's C1(T) being a parametric curve defined by means of 3 real functions of T (T [0,1]) X1(T) [0,1], Y1(T) [0,1] and Z1(T) [0,1] such as :
X1(T=0)=0 Y1(T=0)=0 Z1(T=0)=0 (lower left foreground corner)
X1(T=1)=1 Y1(T=1)=0 Z1(T=1)=0 (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)}  [0,1]x[0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T),Zi+1(T)}  [0,1]x[0,1]x[0,1]

if T  [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  [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  [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  [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  [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  [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  [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  [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 :

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

Here are some examples of Hilbert-like tridimensional curves using different generating curves and in particular "open" knots :

[Plus d'informations à propos des Courbes de Peano et des Nœuds Infinis -en français/in french-]

(CMAP28 WWW site: this page was created on 04/17/2012 and last updated on 06/24/2023 14:40:11 -CEST-)

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