Download the .ipynb file for this notebook, and place your solutions where indicated (you can make more cells for each problem), keeping the original problem descriptions. Upload only one file, which contains all your work; it should be named "HW08_firstname_lastname". Please include comments in your code; this can also help you get partial credit if your code doesn't work.
Then upload it to Blackboard under the Assignments tab. See Collaboration Policy in Homework section of course webpage (it's the same as it was for previous homeworks).
In class we wrote code to draw the inner part of the Apollonian gasket between three mutually externally tangent circles. We tested it with three mutually externally tangent circles of equal radii. Draw (several iterations of) the inner part of the gasket starting with mutually tangent tangent circles of radii $1/2, 1/2, 1/3$.
# Your solution goes here
Modify the code we wrote for the gasket so that it prints out a list of the radii of all the circles in the gasket. Compute this list for the gasket we did in class (starting with three circles of radius $1/2$), as well as for the one in Problem 1 (radii $1/2,1/2,1/3$). What pattern do you notice concerning the list of radii for the second gasket?
(Hint: you may want to look at 1/radius instead of radius).
Extra credit: Rigorously prove that this pattern holds.
# Your solution goes here
Unlike some of the other fractals we drew, for the Apollonan gasket, the sizes of the objects (in this case, circles) added at the $n$th iteration are not all the same. Compute the smallest and largest radii among all the circles added at the last iteration in ap_gasket(c1,c2,c3,iter), for the two configurations of starting circles c1,c2,c3 given in Problem 2, and iter=3,4,5. How do you think the ratio of these two radii behaves as iter tends to infinity?
# Your solution goes here
Play around with Julia sets for different quadratic polynomials $f(z) = z^2 + c$ (remember that for our program to work correctly, you should choose $|c|<2$). Find a Julia set that you find particularly interesting and draw a detailed picture of it.
# Your solution goes here