A Tridimensional Hilbert-like Curve defined with {X1(...),Y1(...),Z1(...)} and based on an 'open' 7-foil torus knot -iteration 1- [Une courbe tridimensionnelle du type Hilbert définie avec {X1(...),Y1(...),Z1(...)} et basé sur un nœud '7-trèfle' torique 'ouvert' -itération 1-].
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)
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
in order to understand the geometrical meaning of the 8 transformations 
and of their order 
.
[See the used color set to display the parameter T]
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)
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
in order to understand the geometrical meaning of the 4 transformations 
and of their order 
.
[See the used color set to display the parameter T]
 | ==> [iteration 11] |  |
 | ==> [iteration 10] |  |
 | ==> [iteration 9] |  |
 | ==> [iteration 10] |  |
See the used color set to display the parameter T.
See the used color set to display the pi digits.
See the used color set to display the parameter T.
See the used color set to display the pi digits.