Homework 1:

1) Towers of Hanoi

a) Obtain a general expression for the minimum number of moves M as a function of the number of discs N.

b) Assuming one second per move, calculate the time T in seconds required to complete the N=64 problem ("Tower of Brahma Legend"), and compare it to the age of the universe (about 15 billion years).

c) What N-tower could a person complete working 8 hours a day for 50 years?

http://www.sewanee.edu/physics/bridge/GLEICK.html

3) Play the following "line chaos game": You need a coin , paper and pencil.

Draw a segment across the page; we will call its length "1". The left end is called H (for Heads) and the right is called T (for Tails).

Rules: Pick any point in the segment (called Xo, the "seed"). Flip the coin. If it turns up Heads then move the point toward the H end so that its distance to H is 1/3 of the previous distance. Conversely, if it is Tails then move the point toward the T end so that its distance to T is 1/3 of the previous distance. Repeat...

As an exercise to practice the game, do the following sequence: Start at the mid-point (Xo=1/2) and sketch the orbit for this succesion of coin flips: H,H,T. You should get something like this:

Your job is to find out what object would emerge if you would play this game for thousands of flips (that is, regardless of the seed you start with, as long as you erase the first few points, and the particular successions that might occur, is there a PATTERN that emerges? Is this an object you know?). A bit of thinking, and a short VPython program might help you.

4) Draw by hand the first three iterations of the fractals generated by removal according to the following rules:

a) "FRACTAL +": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the four corner squares (to get the + shape). Repeat for each remaining square.

b) "FRACTAL H": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the top and bottom mid- squares (to get the H shape). Repeat for each remaining square.

c) "FRACTAL X": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the top, bottom, left and right mid- squares (to get the X shape). Repeat for each remaining square.

d) "FRACTAL O": Start with a square of side length "1". Divide into 9 equal squares (side 1/3 each), and then remove the center square (to get the square O shape). Repeat for each remaining square. What is the "technical" name of this fractal?

Can you calculate the fractal dimension of the above objects?