A bidimensional Sierpinski 'carpet' computed by means of an 'Iterated Function System' -IFS- [Un 'tapis' de Sierpinski bidimensionnel obtenu à l'aide de la méthode des 'Iterated Function Systems' -IFS-].




This bidimensional Sierpinski "carpet" was computed starting with the two following points:
                    A = {0,0,0} (displayed as a bigger Red sphere)
                    B = {1,0,0} (displayed as a bigger Green sphere)
(by the way, one point is enough for this iterative process...). Then, the coordinates of these two points are iteratively transformed using one of the three following linear transformations chosen randomly (each one with a probability equals to 1/3) at each step:
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   |  0  |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + |  0  |     probability=1/3
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0   0  | | Z(i) |   |  0  |
                    \        /   \             / \      /   \     /
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   |  1  |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + |  0  |     probability=1/3
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0   0  | | Z(i) |   |  0  |
                    \        /   \             / \      /   \     /
                    /        \   /             \ /      \   /     \
                    | X(i+1) |   | 1/2  0   0  | | X(i) |   | 1/2 |
                    |        |   |             | |      |   |     |
                    | Y(i+1) | = |  0  1/2  0  |.| Y(i) | + | 1/2 |     probability=1/3
                    |        |   |             | |      |   |     |
                    | Z(i+1) |   |  0   0   0  | | Z(i) |   |  0  |
                    \        /   \             / \      /   \     /
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 and Green for B).


See the bidimensional Sierpinski "carpet" with one starting point and the display of the number of iteration:




See the tridimensional Sierpinski "carpet":




(CMAP28 WWW site: this page was created on 06/14/2005 and last updated on 01/28/2014 15:47:50 -CET-)



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