WittSem 100L: Patterns in Nature

Assignment 6 for Tuesday, October 9, 2007

0. Reading:

Peterson (The Mathematical Tourist), pp. 123-132 (“Packing it in”)

Bring answers (to discuss, not to hand in) to the following questions... or if you have a hard time answering them from the reading, at least identify the part(s) of the reading that addresses them.

Why isn't the image of a fractal that appears on a computer screen a true fractal?

What's a random walk, and how is this used in creating fractal landscapes?

Why is it hard to generate realistic fractal images of trees?

What are some advantages of using fractals to represent natural forms?

How does Barnsley's technique for creating fractals (p. 130) work?

To investigate Barnsley's technique, go to

http://serendip.brynmawr.edu/playground/sierpinski.html

Read the text and try the applet.

Why is randomness important in creating the pattern?

(We'll explore why this technique works in class on Tuesday.)

See other side for problems to hand in!
To hand in (on a separate sheet of paper):

1. Fractal dimension by box-counting:

Go to http://classes.yale.edu/fractals/

From there, go to 13.A. Java Software, and open the Box Counting Dimension software used in class (choose the Java 1.1+ version).

Choose either the Norway or the Chesapeake coastline map.

Find the box-counting dimension of the coastline, using the same method we did in class: count boxes (at least down to size 8—don't forget that you can zoom in), create an Excel spreadsheet with log(1/r) and log(N), make a graph of log(N) vs log(1/r), find the equation of a trendline that fits the data, and from that find a numerical result for d.

Comment on whether your answer makes sense, given what you know about the fractal dimension of various curves and the meaning of the fractal dimension.

2. List the IFS transformations used to create each fractal shown below, in the same format as you would enter them into the Deterministic IFS program. Also briefly explain in words what each transformation does. (You may use the IFS program to check your work; note that the figures below are shown at the 6^th generation.)

a) An upside-down Sierpinski gasket (No rotations or reflections are needed for this one)

b) A relative of the Sierpinski gasket (a coral reef?) Hint: there will be one reflection.