Nature’s Monte Carlo Bakery:
The Story of Life as a Complex System
GEK1530
Frederick H. Willeboordse
[email protected]
1
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?
2
GEK1530
The Bakery
Flour
Water
Get some units
- ergo building blocks
Add
Ingredients
mix n bake
Get something
wonderful!
Process
Knead
Yeast
Wait
Bake
Eat & Live
3
GEK1530
Today's Lecture
Fractals
Is there a geometric analog
to chaos?
Cellular Automata
What is the simplest system
that can display complex
behavior?
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.
4
GEK1530
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).
5
GEK1530
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, Russia
Died: 6 Jan 1918 in Halle, Germany
6
GEK1530
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
7
GEK1530
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.
8
GEK1530
Fractals
Mandelbrot & England
How long is the cost line of England
9
GEK1530
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:
z n 1  z n2  c
Note: The actual Mandelbrot set
are just the black points in the
middle!
All the colored points escape
(but after different numbers of
iterations).
10
GEK1530
Fractals
The Mandelbrot Set
Does this look like the logistic map? It should!!!
z n 1  z n2  c
1
1 2
Take z to be real, divide both sides by c:
z n 1  z n  1
c
c
1
2
x

cx
then substitute xn  z n to obtain n 1
n 1
c
Define:   c And we find the logistic map from before
x n 1  1  αx 2n
11
GEK1530
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.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.
12
GEK1530
Fractals
The Mandelbrot Set
Next, zoom into
this Area.
13
GEK1530
Fractals
The Mandelbrot Set
Next, zoom into this Area.
14
GEK1530
Fractals
The Mandelbrot Set
Next, zoom into this Area.
15
GEK1530
Fractals
The Mandelbrot Set
16
GEK1530
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
17
GEK1530
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.
18
GEK1530
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 …
19
GEK1530
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.
Line
Rule
Time 0
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.
This is how the Cellular
Automaton evolves
Time 1
Time 2
20
GEK1530
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.
Line
Rule
Time 0
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.
This is how the Cellular
Automaton evolves
Time 1
Time 2
21
GEK1530
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.
22
GEK1530
Cellular Automata
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).
23
GEK1530
Cellular Automata
Rule 254:
Rule 254:
We can of course apply this rule to the initial condition we
had before but what to do at the boundary?
24
GEK1530
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).
254:
Now, let us apply rule 254. This is quite simple,
everything, except for three neighboring grey cells will
lead to a black cell.
25
GEK1530
Cellular Automata
254:
Continuing the procedure:
Time 0
Time 1
Time 2
Time 3
26
GEK1530
Cellular Automata
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.
27
GEK1530
Cellular Automata
So let us look at another rule. This one is called rule 90.
Rule 90:
That doesn’t look like it’s very exciting either. What’s the
big deal?
28
GEK1530
Cellular Automata
Applying rule 90.
90:
After one time step:
After two time steps:
At least it seems to be a bit less boring than before….
29
GEK1530
Cellular Automata
Applying rule 90.
90:
After three time steps:
Hey! This is becoming more fun….
30
GEK1530
Cellular Automata
Applying rule 90.
90:
After four time steps:
Hmmmm
31
GEK1530
Cellular Automata
Applying rule 90.
90:
After five time steps:
It’s a Pac Man!
32
GEK1530
Cellular Automata
Applying rule 90.
90:
Well not really. It’s a Sierpinsky gasket:
Which is a fractal!
33
GEK1530
Cellular Automata
Applying rule 90.
90:
Well not really. It’s a Sierpinsky gasket:
From S. Wolfram:
A new kind of
Science.
34
GEK1530
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:
Rule 30
Note that I’ve
only changed the
color of two
boxes compared
to rule 90.
35
GEK1530
Cellular Automata
Applying rule 30.
30:
36
GEK1530
Cellular Automata
Applying rule 30.
30:
While one side has
repetitive patterns,
the other side
appears random.
From S. Wolfram:
A new kind of
Science.
37
GEK1530
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.
38
GEK1530
Cellular Automata
Value 1
Value 2
Value 4
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 1
Value 2
Value 4
=4
=3
Good. Now we know how to get the sequence on the top.
39
GEK1530
Cellular Automata
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
40
GEK1530
Cellular Automata
Like this we can number all the possible 256 rules for this type
of cellular automaton.
41
GEK1530
Cellular Automata
Like this we can number all the possible 256 rules for this type
of cellular automaton.
42
GEK1530
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 …
43
GEK1530
Wrapping up
Key Points of the Day
Simple dynamical rules can
lead to complex behavior
Simple geometric rules can
lead to complex structures
90
:
Cellular automaton rule
References
Fractal
Give it some thought
Can you think of
any ‘real-life’
cellular automata?
44