Impossible Structures






Jean-François COLONNA
[Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
france telecom, France Telecom R&D

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[The Y2K Bug [Le bug de l'an 2000]]
[Real Numbers don't exist in Computers and Floating Point Computations aren't safe. [Les Nombres Réels n'existent dans les Ordinateurs et les Calculs Flottants ne sont pas sûrs.]]
[Please, visit A Virtual Machine for Exploring Space-Time and Beyond, the place where you can find thousands of pictures and animations between Art and Science]
(CMAP28 WWW site: this page was created on 05/12/1998 and last updated on 11/14/2023 17:54:35 -CET-)



[en français/in french]


Abstract: How to build impossible structures?


Keywords: Picture Synthesis, Impossible Structures.



One simple way to build impossible structures is to define a library of elementary tridimensional symbols:


Some elementary symbols used to built impossible structures.



Then, their arbitrary bidimensional assembly will reveal impossibilities:


Periodical impossible structure.
Periodical impossible structure.
Periodical impossible structure.
Periodical impossible structure.
Periodical impossible structure.



Then, colors can be added:


Impossible structure.
Impossible structure.
Impossible structure.

Periodical impossible structure.
Maurits Cornelis Escher meets Piet Mondrian.

My own triple impossible staircase -a Tribute to Maurits Cornelis Escher-.
My own quadruple impossible staircase -a Tribute to Maurits Cornelis Escher-.
Quadruple impossible staircase built by means of a paradoxal structure -a Tribute to Maurits Cornelis Escher-.
Triple impossible staircase and a paradoxal structure.
Triple impossible staircase painting a paradoxal structure.


and/or transformations can be made:


Autostereogram -using a periodical paradoxal structure as a desguise texture- with an hidden gaussian mountain.

Impossible structure.

Paradoxal structure.
Paradoxal structure.
Paradoxal black hole.
Paradoxal structure.

Paradoxal structure -from Order to Disorder-.
Paradoxal structure.

Paradoxal structure in the sky.

Paradoxal structure based on the geometry of the sphere.
Paradoxal structure based on the geometry of the Bonan-Jeener-Klein triple bottle.
Paradoxal structure based on the geometry of the Boy surface.

Paradoxal structure based on the geometry of the sphere.
Paradoxal structure based on the geometry of the Bonan-Jeener-Klein triple bottle.
Paradoxal structure based on the geometry of the Boy surface.

Paradoxal structure based on the geometry of the plane.
Paradoxal structure based on the geometry of the sphere.
Paradoxal structure based on the geometry of the Bonan-Jeener-Klein triple bottle.
Paradoxal structure based on the geometry of the Boy surface.

The Klein bottle described by means of a Bidimensional Hilbert-like Curve -iteration 5-.




Finally, here are some of my oldest impossible structures:


Untitled 0032.
Untitled 0079 (1972-1975).



Copyright © Jean-François COLONNA, 1998-2023.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 1998-2023.