Fractals Joceline Lega Department of Mathematics University of Arizona

Outline  Mathematical fractals Julia sets Self-similarity Fractal dimension Diffusion-limited aggregation  Fractals and self-similar objects in nature  Fractals in man-made constructs  Aesthetical properties of fractals  Conclusion

Julia sets  The pictures on the left represent a (rotated) Julia set.  Consider iterations of the transformation defined in the plane by where c r and c i are parameters.  Example: c r = 0, c i = 1

Julia sets (continued)  More concisely,.  Choose a pair of parameters (c r,c i ).  If iterates of the point z 0 with coordinates (x 0,y 0 ) remain bounded, then this point is part of the corresponding Julia set.  If not, one can use a color scheme to indicate how fast iterates of (x 0,y 0 ) go to infinity.  Here, the darker the tone of red, the faster iterates go to infinity.

Douady's rabbit fractal  The fractal shown below is the Julia set for c r = , c i = 0.745

Self-similarity

Siegel disk fractal  The fractal shown below is the Julia set for c r = c=-0.391, c i =

Julia sets  As one varies (c r,c i ), the “complexity” of the corresponding Julia set changes as well.  The movie shows the Julia sets for c i = 0.534, and c r varying between 0.4 and 0.6.  One would like to measure the “level of complexity” of each Julia set.

Fractal dimension  Consider an object on the plane and cover it with squares of side length L.  Call N(L) the number of squares needed.  For a smooth curve, N(L) ~ 1/L = L -1 and.  For a fractal curve, the fractal dimension is such that d f > 1.

Fractal dimension of Julia sets  D. Ruelle showed that the fractal dimension of the Julia set of the quadratic map is where c = c r + i c i, |c| 2 =c r 2 +c i 2.  In the movie shown before, |c| 2 =c r , with 0.4 ≤ c r ≤0.6.  The fractal dimension measures the “level of complexity” of the fractal.

Diffusion-limited aggregation  Place a seed (black dot) in the plane.  Release particles which perform a random walk.  If the particle touches, the seed, it sticks to it and a new particle is released.  If a particle wanders off the box, it is eliminated and a new particle is released.

Conclusions  Fractals are mathematical objects which are self-similar at all scales.  One way of characterizing them is to measure their fractal dimension.  Many objects found in nature are self- similar, and the fractal dimension of landscape features is close to 1.3.  The human eye appears to be tuned so that objects with a fractal dimension close to 1.3 are aesthetically pleasing.  Such ideas can be used to create fractal- based virtual landscapes.

Virtual landscape Created with Terragen™

Examples of research projects  Exploring Julia sets Complex variables (MATH 421, 424) Proof course (MATH 322) MATLAB  Understanding DLA Probability (MATH 464) MATLAB  Applications to bacterial colonies and other growth models ODEs (MATH 454) and PDEs (MATH 456) MATLAB Numerical Analysis (MATH 475)  Exploring self-similarity in nature and in the laboratory.

Homework problems  Julia sets How would you set up a computer program to plot Julia sets? Use MATLAB to set up such a code.  DLA Design a computer code that simulates the random walk of a particle. The simulation should stop if the particle leaves the box or reaches a pre- defined cluster inside the box. Program this in MATLAB.