A pyramidal Menger sponge computed by means of an 'Iterated Function System' -IFS- [Une éponge pyramidale de Menger obtenue à l'aide de la méthode des 'Iterated Function Systems' -IFS-].




This pyramidal Menger sponge was computed starting with the three following points:
                    A = {0,0,0}   (displayed as a bigger Red sphere)
                    B = {1,0,0}   (displayed as a bigger Green sphere)
                    C = {1/2,0,1} (displayed as a bigger Blue sphere)
(by the way, one point is enough for this iterative process...). Then, the coordinates of these three points are iteratively transformed using one of the four following linear transformations chosen randomly (each one with a probability equals to 1/4) at each step:
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   |  0  |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + |  0  |     probability=1/4
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0  1/2 | | Z(i) |   |  0  |
                    \        /   \             / \      /   \     /
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   |  1  |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + |  0  |     probability=1/4
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0  1/2 | | Z(i) |   |  0  |
                    \        /   \             / \      /   \     /
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   |  0  |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + |  0  |     probability=1/4
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0  1/2 | | Z(i) |   |  1  |
                    \        /   \             / \      /   \     /
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   | 1/2 |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + | 1/2 |     probability=1/4
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0  1/2 | | Z(i) |   | 1/2 |
                    \        /   \             / \      /   \     /
Each point {X(i+1),Y(i+1),Z(i+1)} is displayed as a little sphere having the color of the initial point {X(0),Y(0),Z(0)} (Red for A, Green for B and Blue for C).


See the pyramidal Menger sponge with one starting point and the display of the number of iteration:




See some artistic views:




See the Sierpinski carpet:




(CMAP28 WWW site: this page was created on 06/14/2005 and last updated on 04/26/2015 11:40:58 -CEST-)



[See all related pictures (including this one) [Voir toutes les images associées (incluant celle-ci)]]

[Please visit the related DeterministicChaos picture gallery [Visitez la galerie d'images DeterministicChaos associée]]
[Please visit the related DeterministicFractalGeometry picture gallery [Visitez la galerie d'images DeterministicFractalGeometry associée]]
[Please visit the related ImagesDesMathematiques picture gallery [Visitez la galerie d'images ImagesDesMathematiques associée]]
[Please visit the related NonDeterministicFractalGeometryNaturalPhenomenonSynthesis picture gallery [Visitez la galerie d'images NonDeterministicFractalGeometryNaturalPhenomenonSynthesis associée]]
[Go back to AVirtualSpaceTimeTravelMachine [Retour à AVirtualSpaceTimeTravelMachine]]
[The Y2K bug [Le bug de l'an 2000]]

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[Mail [Courrier]]
[About Pictures and Animations [A Propos des Images et des Animations]]


Copyright (c) Jean-François Colonna, 2005-2015.
Copyright (c) France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2005-2015.