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