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

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[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.