FractalsComplex Adaptive SystemsProfessor Melanie Moses

March 31 2008

– No office hours today– No class Monday 4/14– Reading for Wednesday: Flake chapter 6– Reading for next week: Fractals in Biology

• A general model of allometric growth (West, Brown & Enquist, Science 1997)

– Assignment 2 due Sunday 4/6– Assignment 1 hardcopies due in class Monday 4/7

How do complex adaptive systems grow?– Example 1: Population growth

• Logistic equation & chaotic dynamics• SIR models

– Example 2: Organism Growth• Fractal networks (L systems)

– Robust– Simple to encode– Growth process is infinite– Can alter (maximize) surface to volume, or area to length ratios

Fractals

Fractals

• Self similarity across scales– The parts look similar to the whole– Can exist in time or in space

• Fractional Dimension– D = 1.5, more than a line, less than a plane– Generated by recursive (deterministic or probabilistic)

processes

The Cantor Set

• Draw a line on the interval [0,1]• Recursively remove the middle third of each line• Algorithmic mapping from the Cantor set end points to natural numbers

– Ternary numbers: 1/3 = 3-1 = 0.1, 2/9 = 2*3-2 = .02, etc.

• Cantor set has an uncountably infinite number of points• At step n, 2n segments, each 1/(3n) wide: • measure of the set at step n is (2/3)n

• Infinitely many pointswith no measure

The Koch Curve

• Draw a line• Recursively remove the middle third of each line• Replace with 2 lines of the same length to complete an equilateral triangle• A curve of corners • Length: 4n line segments, each length 3-n = (4/3)n

• Recursive growth: each step replaces a line with one 4/3 as long• Koch snowflake increases length faster than increasing area: finite area,

infinite length

Fractional Dimension

The length of the coastline increases as the length of the measuring stick decreases(This is strange)

Log

rule

r len

gth

Log object length

Slope = fractionaldimension

Ruler length a, Number N, measure Ma1 = 1m, N1 = 6, M = 6m a2 = 2m, N1 = 3, M = 6m

Flake: N = (1/a)D (proportional to, not =) D = log N / log(1/a)

€

N∝ (1

a)D

D =

logN1

N2

⎛

⎝ ⎜

⎞

⎠ ⎟

loga2

a1

⎛

⎝ ⎜

⎞

⎠ ⎟ Lo

g (a

)-1

Log N

Slope = fractionaldimension

a1 = 1m, N1 = 36 boxes, each 1m2 M =36m2

a2 = 2m, N2 = 9 boxes, each 4m2, M =36m2

log(36/9)/log(2) = 2

• The length of the Koch curve depends on the length of the ruler – a = 1/3, N = 4, L = 4/3– a = 1/9, N = 16, L = 16/9

• Fractals measure length including complexity• N = (1/a)D

• D = log (N)/ log (1/a)• Cantor set: D = log(2n)/log(3n) = log 2/log 3 = .631 (between 0 and 1)• Koch curve:

D from length of measuring unit vsD from box counting method

D = log (N)/ log (1/a)N is # of segmentsa is ruler length=log(36/16)/log(2)=1.17

D = log(N)/log(1/a)N is # of boxesa is box length=log(260/116)/log(2)= 1.16

L(z) = A(z)/z1-D

where L(z) is the mean tube length at the zth generation and A(z) is a constant function

L-systems and fractal growth

Axiom: BRules: B -> F[-B]+B

F -> FF

F: Draw ForwardG: Go forward fixed length+ turn right- turn left[ save position] remove position| go forward distance computed by depth

Axiom: BRules: B -> F[-B]+B

F -> FF