Homework 2



Homework 2: Due Friday 28 September


Not yet complete: more to be added shortly

  1. Calculate the self-similarity dimension of the following objects. Be sure to sketch the object and here is a nice discussion of self-similarity dimension that may prove useful. And here is another nice page about self-similarity and dimension.
    1. The Sierpinski carpet.
    2. The Cantor middle-fifths set. (I.e., at each stage of the construction, remove the middle fifth of the unit interval.)
    3. The Sierpinski Sponge.

  2. The Koch Curve: You can iteratively build the Koch Curve by using the java applet found here. Take the unit interval to be the zeroth generation.
    1. Write expressions for the number of segments and the total length of the Koch curve at generation n.
    2. Determine the self-similarity dimension of the Koch curve.
    3. In the limit that the generation n goes to infinity, what is the total length of the Koch curve?
    4. Now consider the Koch curve as a function; the height of the curve at a given x is f(x). Is this function continuous? Where is this function differentiable?
    5. Suppose now that you started not with a line, but with an equilateral triangle, and then built a snowflake. I.e., the shape is now a closed curve. As a function of n, determine the perimeter and the area of the snowflake. What happens as n goes to infinity? Discuss.

  3. Find a big map of the coast of Maine. (There's a nice map outside Craig Greene's office.) Measure the length of the coast using successively smaller "sticks" of length epsilon. Use the relation between total length and epsilon to estimate the box-counting dimension. Your estimate should also include an estimate of your error. This web site might be helpful.(20 points)

  4. (This problem might get delayed a week; it depends how far we get this week.) Write the following number in binary and ternary (i.e., in base 2 and in base 3):
    1. 15
    2. 81
    3. .4

  5. What do you think you might do your project on? Write a few paragraphs. If you're undecided, you should write about things you're considering, or non-math/physics things you might want to try applying math to.


[Dave] [Chaos and Complex Systems] [Homework Page] [COA]

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