Friday, February 1, 2013


I made a program using java Swing that can easily draw fractals that are generated as the attractors of the math IFS ( Iterated Function Systems). The following is a sample picture. It is a tile of a self-affine tiling of the plane R^2. The IFS consists of 3 functions {f1(x),f2(x), f3(x)}. The 3 functions are defined as follows:
f1(x) = (B^2)^(-1)(x)
f2(x) = (B*w*B)^(-1)(x + B*h)
f3(x) = B^(-1)(x + d)
where B is the matrix {{1, -1},{1,1}}, w is the matrix {{0,-1},{1,0}}, h = {0,1}, and d={1,0}.

Note that the picture consists of 3 pieces that are similar to each other. The 3 pieces do not overlap. They only touch others at the borders. Though the boundary of the whole picture is very complex, we can use similar copies of this shape of different sizes to tile the whole plane without overlapping! And we only need finitely many different sizes of this shape for the tiling.