GEK1530

1
Natures Monte Carlo BakeryThe Story of Life as
a Complex System

• GEK1530

Frederick H. Willeboordse frederik_at_chaos.nus.edu.s
g
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Cellular Automata Fractals
What could be the simplest systems capable of
wide-ranging or even universal computation? Could
it be simpler than a simple cell?

• Lecture 6

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The Bakery
Water
Yeast
Flour
Get some units - ergo building blocks
AddIngredients
mix n bake
Get something wonderful!
Process
Knead
Wait
Eat Live
Bake
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Today's Lecture
The Story
Fractals
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.
Is there a geometric analog to chaos?
Cellular Automata
What is the simplest system that can display
complex behavior?
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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. (well get back to what this means a little
later).
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Cantor
Cantor was one of the most important
Mathematicians of the late 19th century.
Unfortunately, vigorous opposition to his ideas
contributed to a nervous breakdown and he died in
a mental institution.
Georg Ferdinand Ludwig Philipp Cantor
Born 3 March 1845 in St Petersburg, RussiaDied
6 Jan 1918 in Halle, Germany
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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 ofthis construction.
Remove middle third
And so on ad infinitum
Then remove middle third of what remains
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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.
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Fractals
Mandelbrot England
How long is the cost line of England
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Fractals
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 z0 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).
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Fractals
The Mandelbrot Set
Does this look like the logistic map? It should!!!
Take z to be real, divide both sides by c
then substitute
to obtain
And we find the logistic map from before
Define
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Fractals
The Mandelbrot Set
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.geoci
ties.com/CapeCanaveral/2854/
What Id 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.
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Fractals
The Mandelbrot Set
Next, zoom into this Area.
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Fractals
The Mandelbrot Set
Next, zoom into this Area.
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Fractals
The Mandelbrot Set
Next, zoom into this Area.
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Fractals
The Mandelbrot Set
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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.
Roughly
A fractal is a geometric object
Chaos is a dynamical attribute
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Cellular Automata
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.
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Cellular Automata
Stephen Wolfram

• 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
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Cellular Automata
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.
The first line is always given. This is what is
called the initial condition.
Line
This rule is trivial. It means black remains
black and grey remains grey.
Rule
This is how the Cellular Automaton evolves
Time 0
Time 1
Time 2
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Cellular Automata
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.
The first line is always given. This is what is
called the initial condition.
Line
Another simple rule. It means black turns into
grey and grey turns into black.
Rule
This is how the Cellular Automaton evolves
Time 0
Time 1
Time 2
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Cellular Automata
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.
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Cellular Automata
The 8 possible combinations
Of course, for each possible combination well
need to state to which color it will lead in the
next time step.
Let us look a a famous rule called rule 254
(well get back to why it has this name later).
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Cellular Automata
Rule 254
We can of course apply this rule to the initial
condition we had before but what to do at the
boundary?
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Cellular Automata
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.
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Cellular Automata
Continuing the procedure
Time 0
Time 1
Time 2
Time 3
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Cellular Automata
Of course we dont really need those arrows and
the time so we might just as well forget about
them to obtain
Nice, but well not very exciting.
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Cellular Automata
So let us look at another rule. This one is
called rule 90.
That doesnt look like its very exciting either.
Whats the big deal?
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Cellular Automata
Applying rule 90.
After one time step
After two time steps
At least it seems to be a bit less boring than
before.
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Cellular Automata
Applying rule 90.
After three time steps
Hey! This is becoming more fun.
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Cellular Automata
Applying rule 90.
After four time steps
Hmmmm
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Cellular Automata
Applying rule 90.
After five time steps
Its a Pac Man!
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Cellular Automata
Applying rule 90.
Well not really. Its a Sierpinsky gasket
Which is a fractal!
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Cellular Automata
Applying rule 90.
Well not really. Its a Sierpinsky gasket
From S. Wolfram A new kind of Science.
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Cellular Automata
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.
Surprisingly they do!
Rule 30
Note that Ive only changed the color of two
boxes compared to rule 90.
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Cellular Automata
Applying rule 30.
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Cellular Automata
Applying rule 30.
While one side has repetitive patterns, the other
side appears random.
From S. Wolfram A new kind of Science.
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Cellular Automata
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.
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Cellular Automata
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.
Value 4
Value 2
Value 1
4
Value 4
Value 2
Value 1
3
Good. Now we know how to get the sequence on the
top.
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Cellular Automata
How about the bottom? We can do exactly the same
thing but since we have 8 boxes on the bottom
its counting from 0 to 255.
Value 128
Value 64
Value 16
Value 32
281664 90
Value 1
Value 8
Value 4
Value 2
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Cellular Automata
Like this we can number all the possible 256
rules for this type of cellular automaton.
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Cellular Automata
Like this we can number all the possible 256
rules for this type of cellular automaton.
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Cellular Automata
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
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Wrapping up
Key Points of the Day
Simple geometric rules can lead to complex
structures
Simple dynamical rules can lead to complex
behavior
Fractal
Cellular automaton rule
Give it some thought
References
Can you think of any real-life cellular
automata?