1. Fractals Joceline Lega Department of Mathematics University of Arizona

2. 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

3. Julia sets • The pictures on the left represent a (rotated) Julia set. • Consider iterations of the transformation defined in the plane bywhere cr and ci are parameters. • Example: cr = 0, ci = 1

4. Julia sets (continued) • More concisely, . • Choose a pair of parameters (cr,ci). • If iterates of the point z0 with coordinates (x0,y0) 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 (x0,y0) go to infinity. • Here, the darker the tone of red, the faster iterates go to infinity.

5. Douady's rabbit fractal • The fractal shown below is the Julia set for cr = -0.123, ci = 0.745

6. Self-similarity

7. Self-similarity

8. Self-similarity

9. Self-similarity

10. Self-similarity

11. Siegel disk fractal • The fractal shown below is the Julia set for cr = c=-0.391, ci = -0.587

12. Julia sets • As one varies (cr,ci), the “complexity” of the corresponding Julia set changes as well. • The movie showsthe Julia sets for ci = 0.534, and crvarying between0.4 and 0.6. • One would like tomeasure the “levelof complexity” of each Julia set.

13. 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 df > 1.

14. Fractal dimension of Julia sets • D. Ruelle showed that the fractal dimension of the Julia set of the quadratic map iswhere c = cr + ici , |c|2=cr2+ci2 . • In the movie shown before, |c|2=cr2 + 0.5342, with 0.4 ≤ cr ≤0.6. • The fractal dimension measures the “level of complexity” of the fractal.

15. Diffusion-limited aggregation • Place a seed (black dot) in the plane. • Release particleswhich perform arandom walk. • If the particletouches, theseed, it sticks toit and a new particleis released. • If a particle wandersoff the box, it is eliminated and a new particle is released.

16. 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.

17. Virtual landscape Created with Terragen™ http://www.planetside.co.uk/terragen/

18. 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.

19. 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.