fractals complex adaptive systems professor melanie moses march 31 2008
Post on 21-Dec-2015
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TRANSCRIPT
– 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, 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