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-].
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-].
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).
