MAT 128B: Project I: Using iteration methods to understand fractal geometry
Wednesday, Feb. 21st
To get full credit show all your work
Names:
[100 pts]
In this project we are going to implement a series of computer programs that use iteration
methods to generate di erent gures on on the plane.
Describe how you are splitting the task with your team members and provide evidence you
are using Github.
Filled Julia sets, Julia sets and Mandelbrot sets
i.- An introduction to fractals Read pages 98, 101 in Greenbaum and Chartier.
Some key concepts that you will need to understand are orbit, Filled Julia sets, Julia
set and Mandelbrot set. An important result states that if the orb(0) is bounded then
the Julia set is connected. Implement program in page 100 to generate the Filled Julia
set in Figure 4.13 and to convince yourself that the Filled Julia set of of (z) = z2 is
the unit disk (denoted by D2).
ii.- Generate (and plot) other examples changing the value of c in the func-
tion (z) = z2 +c. Try examples like c = 0:36 + 0:1i or c = 0:123 0:745i. Describe
what happens whenjzj> 2 or when the initial z0 values (chosen in the program in page
100) are changed.
iii.- Constructing the Julia set. Remember the Julia set is the boundary of the
Filled Julia set. We here are going to give an algorithm to generate the Julia set.
It is called the Inverse Iteration method: Given a complex number z = x + iy =
r(cos + sin ) with r = px2 +y2 and = tan 1yx when x > 0 (adding if x 100
vi.- Coloring divergent orbits Write a program that assigns a color to the divergent
orbits in the Julia set. The coloring is assigned according to the time the orbits take to
diverge (i.e. jzj> 100). Two orbits, one that gets jzj 100 at iteration n and another
at iteration n+ 1 have similar colors.
vii.- Newton’s method in the complex plane Using the algorithms developed
above. Can you write a program that uses the Newton method on the complex plane
(as an iteration method)? Looking at the colored orbits. Can you identify the roots of
the polynomial? Choose examples of the form. (z) = zn 1 for a xed n.
viii.- The Mandelbrot set Generate the Mandelbrot set associated with (z) = z2+c.
This is going to be achieved by changing c. Color the values of c according to the number
of iterations that it took for the orbit of 0 to diverge. This was the cover of the journal
Scienti c American in 1985.
References
Bisoi AK and Mishra J. 2001 On calculation of fractal dimension of images Pattern
Recognition Letters Volume 22, Issues 6{7, Pages 631-637
Devaney RL and Keen L 1988 Chaos an fractals. The mathematics behind the computer
graphics. American Mathematical Society.