1 GEK1530 Frederick H. Willeboordse Nature’s Monte Carlo Bakery: The Story of Life as a Complex System

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?

GEK 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

GEK 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?

GEK 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 (we’ll get back to what this means a little later).

GEK 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.

GEK 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

GEK 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.

GEK Fractals Mandelbrot & England How long is the cost line of England

GEK 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

GEK 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

GEK 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: 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

GEK Next, zoom into this Area. Fractals The Mandelbrot Set

GEK Next, zoom into this Area. Fractals The Mandelbrot Set

GEK Next, zoom into this Area. Fractals The Mandelbrot Set

GEK Fractals The Mandelbrot Set

GEK 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

GEK 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

GEK 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

GEK 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

GEK 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

GEK 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

GEK 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

GEK 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

GEK 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

GEK Continuing the procedure: 254: Time 0 Time 1 Time 2 Time 3 Cellular Automata

GEK 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

GEK 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

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

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

GEK Applying rule : Hmmmm After four time steps: Cellular Automata

GEK Applying rule : It’s a Pac Man! After five time steps: Cellular Automata

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

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

GEK 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

GEK Applying rule : Cellular Automata

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

GEK 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

GEK 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

GEK 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. = = 90 Value 128 Value 64 Value 16 Value 32 Value 8 Value 4 Value 2 Value 1 Cellular Automata

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

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

GEK 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

GEK 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 :