Impossible Structures






Jean-François COLONNA
www.lactamme.polytechnique.fr
jean-francois.colonna@polytechnique.edu
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, 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]]
[Do you believe that Real Numbers exist for a computer and that floating point computations are safe?]
[Please, visit A Virtual Space-Time Travel Machine, the place where you can find more than 6120 pictures between Art and Science]
(CMAP28 WWW site: this page was created on 05/12/1998 and last updated on 11/22/2016 15:52:03 -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
.

Periodical impossible structure
.

Triple impossible staircase built by means of a paradoxal structure -a tribute to M.C. Escher-
.

Double impossible staircase and a paradoxal structure
.

Double 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
.

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's triple Klein 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's triple Klein 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's triple Klein bottle
.

Paradoxal structure based on the geometry of the Boy surface
.




Finally, here are some of my oldest impossible structures:


No Title 0032
.

No Title 0079 (1972-1975)
.



Copyright (c) Jean-François Colonna, 1998-2016.
Copyright (c) France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 1998-2016.