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
One-dimensional Cellular Automata
Time
Transition Rules
Wolfram’s Classification Scheme
• I: Steady state at end
• II Repetitive cycle
• III: Random-like 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: Steady-state Patterns
Type 2: Repetitive Cycles
Type 3: Random-like patterns
Type 4: Local Structures that Move
Langton’s Lambda Parameter
l=10/32, Type II
l=14/32, Type III
l=12/32, Type IV
Rule 30 (Wolfram, 2002)
This rule produces complex patterns with even
the simplest initial condition (one “on” cell)
Sensitivity to initial conditions
Rule 22
Rule 30
Changing one cell in initial seed pattern causes a
cascade of changes
Cellular Automata Terminology
• Cell-space: define a lattice structure with maximum
extent of n columns and m rows
m
L = {(i, j) | i, j Œ N,0 £ i < n,0 £ j < m}
n
• Moore neighborhood: N, S, E, W and diagonal neighbors
†
N i, j = {( k,l) Œ L k - i £ 1and l - j £ 1}
• Von Neumann: only N, S, E, W cells
†
N i, j = {( k,l) Œ L k - i + l - j £ 1}
†
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
• Self-similarity 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
log(N)
An objectDhas N units of measure A
D=
Ê1ˆ
N =Á ˜
Ë A¯
D= dimensionality
Ê1ˆ
logÁ ˜
Ë 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)
T
log
2
Ê1ˆ
(
) = T log(2) @ 0.6309
A = Á T ˜, N = 2T ,D =
Ë3 ¯
log( 3T ) T log(3)
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
2-D substitution systems
L-Systems 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
U[1,6]
1 1 1 1 5 5 5 5 1 1 1
0 0 0 4 0 0 0 -4 0 0
+
0 0 14 -1
-4 0 0 -1
-4 14 0
1 1 2 4 5 4 2
U[2,6]
1 1
1
∂t u[t, x] = ∂ xx u[t, x]
4
dx
dxx
†
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)
Model-world comparison
Plant Formation
Pine Cone Spirals
The numbers of clockwise and counter-clockwise
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
C=1
C
A
A
=
A
C-A
C2-AC=A2
A
C-A
A2+A-1=0
The Golden Section
C=1
A
Find the A such that
C
A
=
A
C-A
C-A
C2-AC=A2
A2+A-1=0
Golden Rectangle
f
1
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 f
f=.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
New seed is positioned maximally as far away from
existing seeds as possible.
2
1
Sunflower Seed Interactions
2
1
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
4
2
≈137.5o
1
3
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
• Activator-inhibitor systems
Cells activate and inhibit neighboring cells
Close neighbors activate each other
Further neighbors inhibit each other
Mexican hat function in vision
Influence on cell
–
–
–
–
Distance from cell
Turing’s Reaction-Diffusion Model
• Show how patterns can emerge through a self-organized
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) Reaction-Diffusion Model
Reaction
Diffusion
A
-
+
B
reaction
diffusionion
A difference equation account of diffusion
a=f(x)
xi-1
xi
xi+1
ai-1
ai
ai+1
Da/Dx
ai- ai-1
D2a/Dx2
ai+1-ai
(ai-1+ai+1)-2ai
2
Da
2 = ai-1 + ai +1 - 2ai
Dx
t +1
t
t
t
t
t
ax,y
= ax,y
+ DtDa ( axt +1,y + ax-1,y
+ ax,y
+
a
4a
+1
x,y-1
x,y )
Pattern Formation with activator-inhibitor 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