Comp 2673
Homework 1
Fractals
You may work with one other person on this assignment
Due Wednesday, April 2

1. Draw the 4th iteration of a Koch's curve. Is this a fractal? To answer this question, see which of the 4 properties of fractals it has.
2. Showing that the length of the Koch's curve is infinite would indicate that its dimension should be more than 1. Show that the length of Koch's curve is indeed infinite (find a formula for the length of the 1st, 2nd, 3rd,...,nth iteration, and take the limit as n goes to infinity).
3. Showing that the area of the Sierpinski gasket is 0 would indicate that its dimension should be less than 2. Show that the area of the Sierpinski gasket is 0. Do this by showing that the area of the removed triangles totals to the area of the original triangle. Do this by calculating the total area removed at the 1st, 2nd, 3rd, ..., nth step, then take the limit as n goes to infinity.
4. Calculate the scaling dimension of the Sierpinski gasket.
5. Here's the description of an iterated function system. The iterated function sytem has 3 transformations:
   • Scale down to .707 of the original size in both directions Place this image rotated 45 degrees clockwise, centered in the original square.
   • Scale down by a factor of 3 in both directions, place the image upside-down with its base touching the bottom of the square, and centered.
   • Scale down by a factor of 3 in each direction, and place it in the lower left-hand corner.
   1. Start with ANY square non-blank image of your choice. Use a scaling photocopy machine, scissors and tape to create a 3rd iteration of the above iterated function system. Turn in the original image, the 1st, 2nd, and 3rd images.
   2. Before you begin, tell me how many photocopies you will have to make to get the final product (if you don't make any mistakes.)
   3. Write down the matrices that describe these 3 linear transformations.
6. Use the IFS from the previous problem. Starting with the point (0,0), find the image point after 3 iterations if you choose function 1, followed by function 3, followed by function 2.