Hilbert Curves and Infinite Knots
Jean-François COLONNA
jean-francois.colonna@polytechnique.edu
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, 91120 Palaiseau, France
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[en français/in french
In the year 1890 Giuseppe Peano discovered curves that are going through all the points of a square (and of a cube) showing then
continuous surjections between [0,1] and [0,1]x[0,1] (and [0,1]x[0,1]x[0,1]).
During the following years, David Hilbert defined more such curves that can be generalized.
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
:
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)=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)} ∈ [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
:
[More information about Peano Curves and Infinite Knots -in english/en anglais-]
[Plus d'informations à propos des Courbes de Peano et des Nœuds Infinis -en français/in french-]
The infinite knots:
Nota:
A mathematical node is a closed curve but to be used as a C1(T)-like curve it has to be open
(for the sake of readability, the two "free" extremities of the Ci(T) curves are not linked:
they are the lower foreground cube vertices, at left and at right respectively displayed as bigger spheres...).
Let's use a 3-foil torus knot 
as the parametric curve C1(T) and let's open it .
Here are the five first curves
Ci(T) (i ∈ [1,5]):
See the used color set to display the parameter T.
After an infinite number of iterations the curve fills the [0,1]x[0,1]x[0,1] cube: it is a first "infinite knot"...
Now, let's use a 5-foil torus knot 
as the parametric curve C1(T) and let's open it .
Here are the five first curves
Ci(T) (i ∈ [1,5])
giving after an infinite number of iterations a second "infinite knot"
:
See the used color set to display the parameter T.
At last, let's use a 7-foil torus knot 
as the parametric curve C1(T) and let's open it .
Here are the five first curves
Ci(T) (i ∈ [1,5])
giving after an infinite number of iterations a third "infinite knot"
:
See the used color set to display the parameter T.
Copyright © Jean-François Colonna, 2022-2023.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2022-2023.