1. Fractals Complex Adaptive Systems Professor Melanie Moses March 31 2008

2. – 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

3. 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

4. Fractals

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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:

12. 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

13. 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

14. L-systems and fractal growth Axiom: B Rules: B -> F[-B]+B F -> FF

15. 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