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-].
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-].
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).
