Cellular Automata

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


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


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