From Squares and Cubes to Surfaces (bidimensional Manifolds) and tridimensional Manifolds

Jean-François COLONNA

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, 91120 Palaiseau, France

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(CMAP28 WWW site: this page was created on 04/17/2023 and last updated on 07/03/2023 18:52:09 -CEST-)

Contents:

- 1-Georg Cantor:
- 2-David Hilbert, Giuseppe Peano and more:
- 3-Tridimensional Surfaces (Bidimensional Manifolds):
- 4-Tridimensional Manifolds:
- 5-Beyond:

Here are the five first bidimensional Hilbert curves with an increasing number of iterations :

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

This process can be generalized, and it is thus possible to fill a cube (in R

Here are the four first tridimensional Hilbert curves with an increasing number of iterations :

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

X = F_{x}(u,v)

Y = F_{y}(u,v)

Z = Fwith:_{z}(u,v)

u ∈ [U_{min},U_{max}]

v ∈ [VFor example:_{min},V_{max}]

F_{x}(u,v) = R.sin(u).cos(v)

F_{y}(u,v) = R.sin(u).sin(v)

Fwith:_{z}(u,v) = R.cos(u)

u ∈ [0,pi]

v ∈ [0,2.pi]defines a sphere with R as the radius and the origin of the coordinates as the center.

[U

v ^ | V |...... --------------------------- max | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| | |+++++++++++++++++++++++++++| V |...... --------------------------- min | : : | : : O-------------------------------------------------> U U u min max

Let's define a curve that fills the [0,1]x[0,1] square :

^ | 1 |--------------- | +++++ +++++ | | + + + + | | + +++++ + | | + + | | +++++ +++++ | | + + | | +++++ +++++ | 0 O------------------------> 0 1

Obviously one can define a mapping between the [0,1]x[0,1] square and the [U

v ^ | V |...... --------------------------- max | | ++++++++ ++++++++ | | | + + + + | | | + + + + | | | + +++++++++++ + | | | + C + | | | ++++++++ ++++++++ | | | + + | | | + + | | | ++++++++ ++++++++ | V |...... --------------------------- min | : : | : : O-------------------------------------------------> U U u min maxThen it suffices to display only the points {F

Here are some examples of this process :

Surface | ==> | C Curve | ==> | Surface filling Curve |

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X = F_{x}(u,v,w)

Y = F_{y}(u,v,w)

Z = Fwith:_{z}(u,v,w)

u ∈ [U_{min},U_{max}]

v ∈ [V_{min},V_{max}]

w ∈ [W[U_{min},W_{max}]

It is obvious to generalize the preceding bidimensional process in the tridimensional space...

Here are some examples of this process :

Tridimensional Manifold | ==> | C Curve | ==> | Tridimensional Manifold filling Curve |

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Here are some examples of these extended processes :

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