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Complex Adaptive Systems, CS 591

Assignment 3: Fractals

Due midnight April 21,2008

For this assignment (worth 50 rather than 100 points) you do not need to prepare a report.Due midnight April 21,2008

You only need to do the following.

Write a program that draws trees following the West et al 1997 algorithm. To make things

easier, draw your trees in 2 dimensions so that each branch has a width and a length (not a

radius). The length ratio (y) is set so that the branches are area filling (as opposed to volume

filling in West et al). Thus, given a branching ratio (n), y= n1/2 rather than y= n1/3. Similarly,

rather than being area-preserving, the widths of branches should be width-preserving, that

so that ft = n1. In the 2-D version, of West et al predict Nt ~A„et?ft, where JVt = the number of

terminal units andA,et = the area (or footprint) of the network.

Allow the user to specify a depth (N) of the tree, the number of daughter branches (n) per

parent and the branching angle (0). Specify the size of the 'invariant terminal unit1 (the leaf

or capillary) to be length ln= 2 and width wn= 1/2, so that the area of a terminal unit An = 1.

Build your tree from the leaves back to the trunk.

For each tree your program should output:

a) A 2D drawing of your tree.

b) Report the number of leaves (Nt), the Area of the whole tree (Ami), and the ratio of

leaf area to whole network area ((Wt* A)/A„et) given N.

c) Calculate and report the fractal dimension of each tree you generate.

Turn ln

i) Your code

11) Drawings of 2 trees generated by your code (specify JV, n and ff).

iii) A table reporting the items in b) and c) for trees with the following inputs

(12 lines in your table): N = 2,4,8,10; n = 2,4,8

iv) A mathematical description of

a. the relationship between Nt and A„ec

b. how fractal dimension depends on n and N?

Extra credit (10 points if you solve how fractal networks fit ln Euclidean organisms, partial

credit for a good effort). Consider that the networks you generated above are circulatory

networks for 2D circular animals (small networks for circular mice, big networks for

circular cows). If the maximum distance between leaves of the tree equals the diameter of

the animal, how big is the animal, i.e. what is its area? (This makes the most sense visually if

you use n=4 in 2d). Calculate the area of the animal (given the diameter above) as a function

of the area of the network.

You should see that the area of the circle does not equal the area of the network. West et al

propose the following (translated to 2D):

a. The number of capillaries is n" and n" oc AnM2/3

b. To achieve A„eta Adrcie, set Nt °cAcircie2/3. then AMt« Nt3/2 \< x Acireie2''3)2/3o<:Acirde.

Can you draw a tree, and an organism surrounding that tree that shows whether these

conditions cause A„et ~ ACMe? Can you relax the assumption that the animal is circular to

show that this is true under other assumptions? Can you relax the assumption that the

terminal units are invariant to solve this problem? Can you systematically change 8 or n as

you change N to cause Anet ~ AcWe? Show your results with fractal branches that fill the area

of the circular organisms, and explain them mathematically.

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