Fractals Complex Adaptive Systems Professor Melanie Moses March

– 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

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, 2 n segments, each 1/(3 n ) wide: measure of the set at step n is (2/3) n Infinitely many points with 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: 4 n 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 ruler length Log object length Slope = fractional dimension

Ruler length a, Number N, measure M a 1 = 1m, N 1 = 6, M = 6m a 2 = 2m, N 1 = 3, M = 6m Flake: N = (1/a) D (proportional to, not =) D = log N / log(1/a) Log (a) -1 Log N Slope = fractional dimension a 1 = 1m, N 1 = 36 boxes, each 1m 2 M =36m 2 a 2 = 2m, N 2 = 9 boxes, each 4m 2, M =36m 2 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(2 n )/log(3 n ) = log 2/log 3 =.631 (between 0 and 1) Koch curve:

D from length of measuring unit vs D from box counting method D = log (N)/ log (1/a) N is # of segments a is ruler length =log(36/16)/log(2) =1.17 D = log(N)/log(1/a) N is # of boxes a is box length =log(260/116)/log(2) = 1.16

L(z) = A(z)/z 1-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: B Rules: B -> F[-B]+B F -> FF

F: Draw Forward G: Go forward fixed length + turn right - turn left [ save position ] remove position | go forward distance computed by depth Axiom: B Rules: B -> F[-B]+B F -> FF