NOTE: This assignments can be done in the lab or in your room.

 

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 computer program might help you... WARNING: Playing the actual game thousands of times is NOT the best way to spend your time in Sewanee!.


1-2) Draw by hand (and if you can, also using TrueBasic) four 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.

b) "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 coastline, lake, country, etc.). Compare with other people's results.