Cellular
Automata
•
Grid of cells, connected to neighbors
–
Spatial
organization.
Typically
1
or
2
dimensional
•
Time and space are both discrete
•
Each
cell
has
a
state
–
Cell’
s
state
at
t+1
depends
only
on
states
of
its
neighbors
and
itself
at
t.
Behavior
is
determined
locally
Onedimensional
Cellular
Automata
Transition Rules
Time
Wolfram’
s
Classification
Scheme
•
I: Steady state at end
•
II
Repetitive
cycle
•
III: Randomlike behavior
–
Rule
30
–
Cannot
compress
behavior
(other
than
by
using
Rule
30)
•
IV: Complex patterns with local structures that move through
space/time
–
Edge
of
Chaos?
(Langton,
Crutchfield,
Kauffman)
–
Langton’
s
Lambda
parameter
•
Number of rules producing a live cell/Total number of rules
–
Not
too
rigid
and
not
too
fluid
–
Information
can
be
effectively
transmitted
Type
1:
Steadystate
Patterns
Type
2:
Repetitive
Cycles
Type
3:
Randomlike
patterns
Type
4:
Local
Structures
that
Move
Langton’
s
Lambda
Parameter
=10/32, Type II
=12/32, Type IV
=14/32, Type III
Rule
30
(Wolfram,
2002)
This
rule
produces
complex
patterns
with
even
the
simplest
initial
condition
(one
“
on”
cell)
Sensitivity
to
initial
conditions
Changing
one
cell
in
initial
seed
pattern
causes
a
cascade
of
changes
Rule 30
Rule 22
Cellular
Automata
Terminology
•
Cellspace: define a lattice structure with maximum
extent of n columns and m rows
•
Moore neighborhood: N, S, E, W and diagonal neighbors
•
Von Neumann: only N, S, E, W cells
L
(
i
,
j
)

i
,
j
N
,
0
i
n
,
0
j
m
N
i
,
j
k
,
l
L
k
i
1
and
l
j
1
N
i
,
j
k
,
l
L
k
i
l
j
1
n
m
Cellular
Automata
Terminology
•
Totalistic rules
–
the state of the next state cell is only dependent upon the sum of the
states of the neighbor cells
•
Reversible rules
–
No application of the rules loses any information
–
For every obtainable state there is only state that can produce it
–
Atypical, because these do not incorporate cell interactions
–
Sometimes applied in modeling physical systems (e.g. billiard balls)
Cellular
Automata
Broadened
•
Mobile automata
–
A single active cell, which updates its position and state
•
Turing Machines
–
The active cell has a state, and states determine which transition rule is
applied
•
Substitution Systems
–
On each iteration, each cells is replaced with a set of cells
•
Tag systems
–
Remove cells from left, and add to the right depending on removed cells
•
Continuous state systems
–
On each iteration, each cells is replaced with a set of cells
•
Asynchronously updating systems
Mobile
automata
Turing
Machines
Substitution
System
Cantor’s
Set
Fractals
•
Selfsimilarity at multiple scales
•
Formed by iteration
•
Fractional dimensionality
–
The Cantor set: replace every 1 pattern with 101 with same length
–
Cantor set = the points remaining 1 when this is applied infinite times
–
Infinite number of points, but no length
–
A = measure of a measuring device
–
An object has N units of measure A
N
1
A
D
D=
dimensionality
D
log(
N
)
log
1
A
If
A=
1/3
and
D
=2,
N=9
If
A=
1/3
and
D
=1,
N=3
Cantor’s
Set
A = 1/3, N= 2, so D=log(2)/log(3)
A = 1/9, N= 4, so D=log(4)/log(9)
A
1
3
T
,
N
2
T
,
D
log
2
T
log
3
T
T
log(
2
)
T
log(
3
)
0.6309
Dimensionality is between 0 and 1
Hilbert’s
Space
Filling
Curve
•
Dimensionality
=
2
as
iterations
go
to
infinity
even
though it is a single line
•
Fractals: measure of object increases as the measuring
device
decreases
2D
substitution
systems
LSystems
for
plant
growth
Substitution system
Continuous
State
Cellular
Automata
•
Each cell’s state is based on a numeric function of
neighbors
–
Diffusion = each
–
cell’s state is average of itself and its 2 neighbors
•
Space,
state,
and
time
can
all
be
continuous
–
Partial differential equations: Specify the rate at which gray levels change with time
at every point in space. Depends on gray level at each point in space, and on the
rate at which gray levels change with position
•
Partial Differential Equation for Diffusion
t
u
[
t
,
x
]
1
4
xx
u
[
t
,
x
]
1 1 1 1 5 5 5 5 1 1 1
0 0 0 4 0 0 0 4 0 0
0
0
4
4
0
0
4
4
0
0
0
1
1
0
0
1
1
0
1 1 2 4 5 4 2 1 1
x
xx
U
[1,6]
U
[2,6]
+
Continuous
States
Diffusion
=
every
cell
takes
on
the
average
of
itself
and
its
two
neighbors
Continuous
States
and
Space
Discrete
transitions
from
continuous
systems
Order from random configurations
Apparent randomness from orderly configurations
Crystal
Formation
When ice added to snowflake, heat is released, which
inhibits the addition of further ice nearby
Cellular automata: cell becomes black if they have exactly
one black neighbor, but stay white if they have more than
one
black
neighbor
Crystal
Formation
Shell
formation
(following
Raup)
Modelworld
comparison
Plant
Formation
Pine
Cone
Spirals
The
numbers
of
clockwise
and
counterclockwise
spirals
are
successive
numbers
in
the
Fibonacci
sequence:
1
1
2
3
5
8
13
21
34
55
The
angle
between
successive
leaves
on
the
pine
cone
is
137.5
degrees
Golden
Mean
A
C=1
A
CA
C
A
=
A
C  A
C
2
AC=A
2
A
2
+A1=0
The
Golden
Section
A
C=1
CA
C
2
AC=A
2
A
2
+A1=0
C
A
=
A
C  A
Find the A such that
1
Golden Rectangle
The
Golden
Section
The
angle
between
successive
sunflower
seeds
is
the
golden section of a circle
The
ratio
of
successive
numbers
of
a
fibonacci
sequence
approximate
=.6180… 3/5=.6 8/13=.615 34/55=.6182
The
Golden
Section
in
Plants
So,
are
sunflowers
good
mathematicians?
No,
137.5
degrees
emerges
from
simple
interactions
among
plant
leaves/seeds
Sunflower
Seed
Interactions
1
Sunflower
Seed
Interactions
1
2
New seed is positioned maximally as far away from
existing seeds as possible.
Sunflower
Seed
Interactions
1
2
3
Seeds 1 and 2 both push Seed 3 away, but Seed 2 pushes more
because it is closer to Seed 3.
Find location on circle for seed that minimizes the sum of the
“push”
exerted by other seeds, where push is an inverse square
function of distance
Sunflower
Seed
Interactions
1
2
3
4
≈
137.5
o
A simple model based on these interactions can
provide an account of many plant forms that are
found by varying only a few parameters.
Goodwin

evolutionary
pressures
as
overrated?
Cellular
Automata
in
Shell
Patterns
Pattern
Formation
Pattern
Formation
(Morphogenesis)
•
Spots and Stripe formation
•
Activatorinhibitor systems
–
Cells activate and inhibit neighboring cells
–
Close neighbors activate each other
–
Further neighbors inhibit each other
–
Mexican hat function in vision
Distance from cell
Influence on cell
Turing’s
ReactionDiffusion
Model
•
Show how patterns can emerge through a selforganized
process from random origins
•
Each cell has two chemicals
–
Chemical A is an autocatalyst  it produces more of itself
–
Chemical B inhibits production of A
•
Diffusion: each chemical spreads out
•
Reaction: each chemical reacts to the presence of the
other chemical and to itself
•
Activator chemical diffuses more slowly than inhibitor
chemical
•
If there is local variation in chemicals and chemical
amounts do not increase without bound, then stable
states of inhibitor and activator chemicals are found
Turing’s
(1952)
ReactionDiffusion
Model
Diffusion
Reaction
reaction
diffusion
ion
A
B

+
A
difference
equation
account
of
diffusion
a=f(x)
a
i
1
a
i
a
i
+1
x
i
x
i+1
x
i1
(a
i1
+a
i+1
)2a
i
2
a
x
2
a
i
1
a
i
1
2
a
i
a
i

a
i
1
a
i
+1

a
i
a/
x
a/
x
2
a
x
,
y
t
1
a
x
,
y
t
tD
a
a
x
1
,
y
t
a
x
1
,
y
t
a
x
,
y
1
t
a
x
,
y
1
t
4
a
x
,
y
t
Pattern
Formation
with
activatorinhibitor
system
Stripe
formation
Greater
diffusion
in
one
direction
than
the
other
Cellular
Automata
for
Animal
Pigmentation
Patterns
Murray
(1993)
Cellular
Automata
for
Animal
Pigmentation
Patterns
Diffusion Limited Aggregation for Population Growth
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