1. 1 GEK1530 Frederick H. Willeboordse frederik@chaos.nus.edu.sg Nature’s Monte Carlo Bakery: The Story of Life as a Complex System

2. 2 Cellular Automata & Fractals Lecture 6 What could be the simplest systems capable of wide-ranging or even universal computation? Could it be simpler than a simple cell?

3. GEK1530 3 The Bakery Add Ingredients Process Flour Water Yeast Knead Wait Bake Get some units - ergo building blocks Eat & Live Get something wonderful! mix n bake

4. GEK1530 4 Today's Lecture The Story The logistic map discussed last time is the best known example for dynamic chaotic behavior. Today we will see that there is something similar in a geometric sense. Then, since we now know that simple systems can behave in unexpected ways, we will explore what is probably the simplest system displaying complex behavior. What is the simplest system that can display complex behavior? Fractals Cellular Automata Is there a geometric analog to chaos?

5. GEK1530 5 Fractals What are Fractals? (roughly) a fractal is a self-similar geometrical object with a fractal dimension. self-similar = when you look at a part, it just looks like the whole. Fractal dimension = the dimension of the object is not an integer like 1 or 2, but something like 0.63. (we’ll get back to what this means a little later).

6. GEK1530 6 Cantor Georg Ferdinand Ludwig Philipp Cantor Born: 3 March 1845 in St Petersburg, Russia Died: 6 Jan 1918 in Halle, Germany Cantor was one of the most important Mathematicians of the late 19 th century. Unfortunately, vigorous opposition to his ideas contributed to a nervous breakdown and he died in a mental institution.

7. GEK1530 7 Fractals The Cantor Set Take a line and remove the middle third, repeat this ad infinitum for the resulting lines. This is the construction of the set! The set itself is the result of this construction. Remove middle third Then remove middle third of what remains And so on ad infinitum

8. GEK1530 8 Mandelbrot Born: 20 Nov 1924 in Warsaw, Poland He discovered what is now called the Mandelbrot set and is responsible for many aspects of fractal geometry.

9. GEK1530 9 Fractals Mandelbrot & England How long is the cost line of England

10. GEK1530 10 The Mandelbrot Set This set is defined as the collection of parameters c in the complex plane that does not lead to an escape to infinity for the equation when starting from z 0 = 0: Note: The actual Mandelbrot set are just the black points in the middle! All the colored points escape (but after different numbers of iterations). Fractals

11. GEK1530 11 Does this look like the logistic map? It should!!! Take z to be real, divide both sides by c: then substitute Define: to obtain And we find the logistic map from before Fractals The Mandelbrot Set

12. GEK1530 12 The Mandelbrot set is strictly speaking not self-similar in the same way as the Cantor set. It is quasi-self-similar (the copies of the whole are not exactly the same). Here are some nice pictures from: http://www.geocities.com/CapeCanaveral/2854/ What I’d like to illustrate here is not so much that fractals can be used to generate beautiful pictures, but that a simple non-linear equation can be incredibly complex. Fractals The Mandelbrot Set

13. GEK1530 13 Next, zoom into this Area. Fractals The Mandelbrot Set

14. GEK1530 14 Next, zoom into this Area. Fractals The Mandelbrot Set

15. GEK1530 15 Next, zoom into this Area. Fractals The Mandelbrot Set

16. GEK1530 16 Fractals The Mandelbrot Set

17. GEK1530 17 Chaos and Fractals How do they relate? Fractals often occur in chaotic systems but the the two are not the same! Neither do they necessarily imply each other. A fractal is a geometric object Roughly: Chaos is a dynamical attribute

18. GEK1530 18 Perhaps one can expect that strange and complex behavior results from very complicated rules. But what are the simplest systems that display complex behavior? This is an important question when we want to figure out whether relatively simple rules could underlie the complexity of life. As it turns out, probably the simplest systems that display complex behaviors are the so-called cellular automata. Cellular Automata

19. GEK1530 19 Born in 1959 in London First paper at age 15 Ph.D. at 20 Youngest recipient of MacArthur ‘young genius’ award Worked at Caltech and Princeton Owner of Mathematica (Wolfram Research) Fantastic publication record … until … 1988 when he stopped publishing in scientific journals From his web site … Stephen Wolfram Cellular Automata

20. GEK1530 20 A (one-dimensional) cellular automaton consists of a line of ‘cells’ (boxes) each with a certain color like e.g. black or grey and a rule on how the colors of the cells change from one time step to the next. Rule Line The first line is always given. This is what is called the ‘initial condition’. This rule is trivial. It means black remains black and grey remains grey. Time 0 Time 1 Time 2 This is how the Cellular Automaton evolves Cellular Automata

21. GEK1530 21 A (one-dimensional) cellular automaton consists of a line of ‘cells’ (boxes) each with a certain color like e.g. black or grey and a rule on how the colors of the cells change from one time step to the next. Rule Line The first line is always given. This is what is called the ‘initial condition’. Another simple rule. It means black turns into grey and grey turns into black. Time 0 Time 1 Time 2 This is how the Cellular Automaton evolves Cellular Automata

22. GEK1530 22 Like this, the rules are a bit boring of course because there is no spatial dependence. That is to say, neighboring cells have no influence. Therefore, let us look at rules that take nearest neighbors into account. or With 3 cells and 2 colors, there are 8 possible combinations. Cellular Automata

23. GEK1530 23 The 8 possible combinations: Of course, for each possible combination we’ll need to state to which color it will lead in the next time step. Let us look a a famous rule called rule 254 (we’ll get back to why it has this name later). Cellular Automata

24. GEK1530 24 Rule 254: We can of course apply this rule to the initial condition we had before but what to do at the boundary? Rule 254: Cellular Automata

25. GEK1530 25 Often one starts with a single black dot and takes all the neighbors on the right and left to be grey (ad infinitum). Now, let us apply rule 254. This is quite simple, everything, except for three neighboring grey cells will lead to a black cell. 254: Cellular Automata

26. GEK1530 26 Continuing the procedure: 254: Time 0 Time 1 Time 2 Time 3 Cellular Automata

27. GEK1530 27 Of course we don’t really need those arrows and the time so we might just as well forget about them to obtain: 254: Nice, but well … not very exciting. Cellular Automata

28. GEK1530 28 So let us look at another rule. This one is called rule 90. That doesn’t look like it’s very exciting either. What’s the big deal? Rule 90: Cellular Automata

29. GEK1530 29 Applying rule 90. 90: At least it seems to be a bit less boring than before…. After one time step: After two time steps: Cellular Automata

30. GEK1530 30 Applying rule 90. 90: Hey! This is becoming more fun…. After three time steps: Cellular Automata

31. GEK1530 31 Applying rule 90. 90: Hmmmm After four time steps: Cellular Automata

32. GEK1530 32 Applying rule 90. 90: It’s a Pac Man! After five time steps: Cellular Automata

33. GEK1530 33 Applying rule 90. 90: Which is a fractal! Well not really. It’s a Sierpinsky gasket: Cellular Automata

34. GEK1530 34 Applying rule 90. 90: Well not really. It’s a Sierpinsky gasket: From S. Wolfram: A new kind of Science. Cellular Automata

35. GEK1530 35 So we have seen that simple cellular automata can display very simple and fractal behavior. Both these patterns are in a sense highly regular. One may wonder now whether ‘irregular’ patterns can also exist. Rule 30: Surprisingly they do! Rule 30 Note that I’ve only changed the color of two boxes compared to rule 90. Cellular Automata

36. GEK1530 36 Applying rule 30. 30: Cellular Automata

37. GEK1530 37 Applying rule 30. 30: While one side has repetitive patterns, the other side appears random. From S. Wolfram: A new kind of Science. Cellular Automata

38. GEK1530 38 Now let us look at the numbering scheme The first thing to notice is that the top is always the same. This is the part that changes. Now if we examine the top more closely, we find that it just is the same pattern sequence that we obtain in binary counting. Cellular Automata

39. GEK1530 39 If we say that black is one and grey is zero, then we can see that the top is just counting from 7 to 0. Good. Now we know how to get the sequence on the top. Value 1 Value 4 Value 2 Value 1 Value 4 Value 2 = 4 = 3 Cellular Automata

40. GEK1530 40 How about the bottom? We can do exactly the same thing but since we have 8 boxes on the bottom it’s counting from 0 to 255. = 2+8+16+64 = 90 Value 128 Value 64 Value 16 Value 32 Value 8 Value 4 Value 2 Value 1 Cellular Automata

41. GEK1530 41 Like this we can number all the possible 256 rules for this type of cellular automaton. Cellular Automata

42. GEK1530 42 Like this we can number all the possible 256 rules for this type of cellular automaton. Cellular Automata

43. GEK1530 43 Like this we can number all the possible 256 rules for this type of cellular automaton. And of course, one does not need to restrict oneself to two colors and two neighbors … Cellular Automata

44. GEK1530 44 Wrapping up Key Points of the Day Give it some thoughtReferences Can you think of any ‘real-life’ cellular automata? Simple dynamical rules can lead to complex behavior Cellular automaton rule Simple geometric rules can lead to complex structures Fractal 90 :