1. Iterated Function Systems (IFS) and Fractals Math 204 Linear Algebra November 16, 2007

2. Affine Transformations Affine transformations F: R 2 => R 2 are mappings made by 2 x 2 matrix multiplication then translation. 2 x 2 matrix multiplication then translation. An affine map is defined by four numbers: a b c d e f Affine maps combine four possible actions on a triangle.

3. Facts about affine maps   The composition of two affine maps is an affine map.   Affine maps can be defined by triangle mappings. That is, if we describe how to map the vertices of a triangle ABC to triangle DEF, this defines an affine mapping.   An affine map F: R 2 => R 2, is called a contraction if for any two points p, q in the plane, the distance between F(p) and F(q) is less than the original distance between p and q.   Given a geometric figure S in the plane, we can ask what “collages” we can make by applying several contraction mappings to S and plotting all these contracted versions of S together.

4. Example of a Fractal Collage: Sierpinski Triangle

5. Affine Contractions Create Collages  Collage Theorem: Given any finite collection of contractions F 1, F 2, …, F n, there is a unique geometric object S such that S = F 1 (S) U F 2 (S) U … U F n (S) S = F 1 (S) U F 2 (S) U … U F n (S)  That is, S is the unique collage defined by these contractions.  FDESIGN is a old DOS program which allows us to easily design collages by drawing triangle mappings then showing us the resulting unique figure that is defined by the mappings.

6. Barnsley Fern showing the four contractions which define this figure

7. Random Iteration Algorithm: How FDESIGN draws pictures  FDESIGN works by keeping track of your affine mappings and assigning a probability p to each one. FractInt IFS format: a b c d e f p  Starting with a random point (x,y) FDESIGN picks one of your affine maps according to its probability, applies it to the point (x,y) to get a new point and repeats (iterates) this action endlessly. Recall this is same as picking a random value 1-3 then shrinking 1/2 way to toward the vertex of our Sierpinski triangle.  Fact: This random iteration algorithm will draw as much of our collage as we wish if we wait long enough.