Necessary geometry
The first step in learning to grow fractal pictures is to understand some of the geometry of plane transformations.
Iterated Function Systems
Now we are ready to learn how to find the rules to generate given fractals. We begin with a right isosceles Sierpinski gasket. Certainly, the gasket can be viewed as made up of three copies of itself, each scaled by a factor of 1/2 in both the x- and y-directions.
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To determine the translation amount of each piece, take some point of the whole fractal (the lower left corner, for example) and observe where that point goes in each piece.
Here we derive the rules for the right isosceles Sierpinski gasket.
Note that applying all three of these transformations to the gasket gives the gasket again. That is, the gasket is invariant under the simultaneous application of these three transformations. In fact, the gasket is the only (compact) shape left invariant. What happens if we apply these transformations to some shape other than the gasket? We won't be left with the same shape. What happens if we apply these transformations to the resulting shape? What happens if we iterate this process? We observe a sequence of pictures that converges to the gasket, independently of the starting shape. With a scanner, some mischief can be achieved.
This convergence occurs in a much more general setting than the gasket rules. Specifically, suppose T1, ..., Tn are contractions, and P0 is any picture. Generate a sequence of pictures
| P1 = T1(P0) U ... U Tn(P0) |
| P2 = T1(P1) U ... U Tn(P1) |
| ... |
| Pk+1 = T1(Pk) U ... U Tn(Pk) |
| ... |
converges to a unique shape, P, the only shape (compact subset of the plane) invariant under the simultaneous application of T1, ..., Tn:
P = T1(P) U ... U Tn(P)
Generating fractals by iterating a collection of transformations is the Iterated Function System (IFS) method. This approach, applying all the transformations to the entire picture, is called the Deterministic Algorithm for generating fractals. Try it here Another approach for generating the images is the Random Algorithm A precursor to the Random Algorithm is the Chaos Game.
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