Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fractals Complex Adaptive Systems Professor 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 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: 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 ruler length Slope = fractional dimension Log object length Ruler length a, Number N, measure M a1 = 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) a1 = 1m, N1 = 36 boxes, each 1m2 M =36m2 a2 = 2m, N2 = 9 boxes, each 4m2, M =36m2 log(36/9)/log(2) = 2 Slope = fractional dimension Log (a)-1 1 N ( ) D a N1 log N 2 D a2 log a1 Log N • 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 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)/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: 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