WittSem 100L: Patterns in Nature

Assignment for Tuesday, October 2, 2007

Hand in your answers for 1-3 on a separate sheet of paper.

1. a) Below is an initiator/generator for a particular fractal curve. Carefully draw the next two stages of the curve using the quadrille-ruled paper provided. Note: all line segments in the generator are the same length.

Initiator (stage 0) generator

b) Assuming that stage 0 has length 1, what is the length of stage 1? What is the length of stage 2?

c) Write an expression for the length at stage N and show that the length goes to infinity as N goes to infinity.

d) What is the fractal (Hausdorff) dimension of this curve?

e) Compare the fractal dimension of this curve to that of the Koch curve. Which is more “space-filling” (that is, closer to a 2D object than a 1D object)? How can you tell?

2. Using the same sort of reasoning that we did in class, show that a straight line has a fractal (Hausdorff) dimension of 1.

3. Run the Sierpinski carpet applet at

http://www.shodor.org/interactivate/activities/SierpinskiCarpet/

a) Figure out and fill in the missing information:

Iteration Number of shaded squares Area of one shaded square Total area

1 1 1 1

2

3

b) How many shaded squares are there at the nth stage? What is the area of one shaded square at the nth stage? What is the total area at the nth stage?

c) What is the limit of the total area as the number of stages approaches infinity?

d) What is the fractal dimension of the Sierpinski carpet (show how you obtained this)?

e) Explain why the fractal dimension makes sense, given what you found for the area of the Sierpinski carpet.

4. Do the reading on the reverse of this page. By midnight on Monday, email me the following:

a) Something you understand better after doing the reading

b) Something you still have questions on or would like to understand better

WittSem 100L: Patterns in Nature

Fractal geometry reading assignment for Tuesday, Oct. 2, 2007

1. Read pp. 114-123 from Peterson, The Mathematical Tourist

2. Read the following sections in the Frame, Mandelbrot, and Neger (FMN) Fractal Geometry website,

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

Instructions: Follow the links in the left frame that are listed in the table below and read the page that comes up. Where it says to click one of the pictures, do so (that usually animates the picture). Where there are other links on a page, follow the ones listed in the table below. Use the browser’s back button to navigate backwards when you’ve reached the end of a “branch.” Feel free to follow links other than those in the table below—this is just the minimum requirement!

Main links in left frame (first level)	Links to follow under main link (second level links)
1.A. Self-similarity	all links
1.B. More examples of self-similarity	Naturalistic fractals
1.C. Initiators and Generators	Sierpinski gasket
	Koch curve
	Cantor set
	Fractal trees
1.D. Geometry of Plane Transformations	scalings, reflections, rotations, translations
2. C. Similarity Dimension	Similarity dimension exercises
2. B. Box-Counting Dimension	boxes line segment square power law box-counting dimension