WEEK 1 (FRACTALS)

1-1) Play the following "line chaos game": You need a coin , paper and pencil.

Draw a segment across the page; we will call its length "1". The left end is called H (for Heads) and the right is called T (for Tails).

Rules: Pick any point in the segment (called Xo, the "seed"). Flip the coin. If it turns up Heads then move the point toward the H end so that its distance to H is 1/3 of the previous distance. Conversely, if it is Tails then move the point toward the T end so that its distance to T is 1/3 of the previous distance. Repeat...

As an exercise to practice the game, do the following sequence: Start at the mid-point (Xo=1/2) and sketch the orbit for this succesion of coin flips: H,H,T. You should get something like this:


Your job is to find out what object would emerge if you would play this game for thousands of flips (that is, regardless of the seed you start with, as long as you erase the first few points, and the particular successions that might occur, is there a PATTERN that emerges? Is this an object you know?). A bit of thinking, and a short VPython program might help you.


1-2) Draw by hand the first three iterations of the fractals generated by removal according to the following rules:

 

a) "FRACTAL +": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the four corner squares (to get the + shape). Repeat for each remaining square.

b) "FRACTAL H": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the top and bottom mid- squares (to get the H shape). Repeat for each remaining square.

c) "FRACTAL X": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the top, bottom, left and right mid- squares (to get the X shape). Repeat for each remaining square.

d) "FRACTAL O": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the center square (to get the square O shape). Repeat for each remaining square. What is the "technical" name of this fractal?

Can you calculate the fractal dimension of the above objects?


1-3) Make a notch in the middle of one side of a piece of typing paper. Hold the sides of the paper firmly and pull them apart (in the plane of the paper) until the paper tears. The tear edge will be jagged and fractal-like. Overlay the torn edge on a piece of graph paper and estimate the dimension of the edge by box counting. Does the dimension depend on the speed with which the paper is torn? Try tissue paper instead of typing paper. Does the character of the edge depend on the structure of the paper?


1-4) Make a copy of fine graph paper on a overhead foil. Use this as one of the sheets in the "blob of goo between two sheets of plastic" experiment: Put a small blob of goo (tootpaste and mud work well) on this sheet, cover with a plain overhead transparency foil, and press firmly on the top foil until the small blob is squeezed out to a large, thin blob. Pull the foils apart, producing a fingery pattern. Do box counting to estimate the average dimension of the boundary of this "dendrite-like" pattern. Try the experiment with different pulling speeds. Does the dimension depend on the speed with which the sheets are separated? Also, try different "goos." Does the dimension depend on the material used?


1-5) Measure the fractal dimension of the boundary of a leaf of your choice. Compare with other people's results.


1-6) Measure the fractal dimension of the boundary of a feature on a map of your choice (could be a costaline, lake, country, etc.. Compare with other people's results.


WEEK 2 (CHAOS)

2-1) Below is a graph of the tent map, defined by the following iteration rule:

 

Xn+1 = s Xn ..........................if Xn<1/2

Xn+1 = s (1-Xn) ....................if Xn>1/2

(s=<2)

Show the maximum height of the tent map graph is s/2.

Show the nonzero fixed point occurs at Xf = s/(1+s).

For s > 1, show the first iterate of s/2 has the value s (1 - s/2 ).

Use graphical iteration to locate four points which the tent map takes to the fixed point w of the Figure below.

Numerically calculate the four values of x you found when s = 2.


2-2) LIFE AND NONLINEAR DYNAMICS

Write a parable of your own. Specifically, take an instance from your life where a small choice, perhaps made in an offhand fashion, led eventually to a significant difference in your life. For your parable to be apt to the context of this course, the train of events flowing from your choice has to be deterministic. Try to identify some of the nonlinearities in the "dynamics of your life" that helped cause your choice to initiate the unexpected chain of events it did.

To emphasize the magnitude of the effect of this small choice, construct a plausible scenario of the result of your making a different small choice (under the same "life dynamics").



2-3) HUMAN FOLLIES

Do some research to find out what do the economists refer to by the "tulips' boom/bust" experience.

What do you think is the connection between market crashes and non-linear systems?

And outbreak of war and non-linear systems?

How about epidemics?


2-4) The Game of Life

 

a) Determine by hand the next generation of the Life configurations below

 a
             
             
             
             
             
 b
             
             
             
             
             
 c
             
             
             
             
             
 d
             
             
             
             
             

 

b) Determine by hand the next two generations of the Life configurations below

First Generations:

 a
             
             
             
             
             
 b
             
             
             
             
             
 c
             
             
             
             
             
 d
             
             
             
             
             

Second Generations:

 a
             
             
             
             
             
 b
             
             
             
             
             
 c
             
             
             
             
             
 d
             
             
             
             
             

 

Using LifeLab and the random initial condition, observe which stable configurations arise spontaneously from the Life rules. (Allow some time to pass before drawing any conclusions - the transients must be allowed to die down.).

A textbook description is no substitute for the experience of Life. Play with LIFE to get a hint of the Game's many wonders. But here's a warning label for the user: Life is powerfully addictive! Use only when not faced with an impending deadline.


2-5) Some deep philosophical questions:

 

"Human consciousness, free will, or feelings can never be understood in terms of physical laws."

"People are mechanistic entities whose behaviors emerge from the same principles that govern all forms of matter."

Here are two opposite views of humanity.

Construct an argument that supports each position.

Which argument do you find more compelling?