Homework due Monday, February 3rd

9. (4 points) Create two different fractals using the two replacement rules seen below. In both cases, start with a straight line segment as the initial shape.

10. (2 points) A fractal that could fit in Lineland is called the Cantor dust. It follows the replacement rule seen below applied to a line segment. In other words, each time the middle third of the line segment is removed.
    • Construct the first four iterates of the Cantor dust.
    • Find the total lengths of all the segments for the first four iterates. (Assume the first segment extends from 0 to 1 on the positive x axis.)
    • Assume you do infinitely many iterates - what do you think the length of the remaining object will be?
    • Name a point that is in the Cantor dust (that is, that is in each iterate.)
    • Why do you think this object is called "dust"?

11. (2 points) Using the website discussed in class, click on the purple object on the side. This will produce the Koch snowflake formed by three copies of the Koch curve.
    • Start by finding the area of the triangle. (Reminder: the area of a triangle is 1/2 the base times the height. You can find the height by using the Pythagorean Theorem. Let a unit length be the distance between dots. Thus the base has a length of 6.)
    • In the first iterate, you add three triangles. Explain why each triangle is one-ninth the size of the original triangle and use this fact to find the area enclosed inside the figure.
    • In the second iterate, you add 12 triangles. Fnd the area of one of them and use that fact to find the total area inside the figure.
    • Give a good argument why the area inside the Koch snowflake is finite (even after infinitely many iterates), but the perimeter is infinite.