305 PHYS - Assignment #2
Due: Friday, January 23rd Make sure your name is listed as a comment at the beginning of all your work. Purpose: Illustrate fractals. Practice programming and plotting.
Mandelbrot Set
The best known fractal and one of the most complex mathematical structure was discovered by Benoit Mandelrot in 1980. It is known as the Mandelbrot Set. The set is produced by the simple map 2 zn+1 = zn + c where the variable z and parameter c are complex numbers and z0 = 0. This relation can be broken in a real and imaginary notation, z = Re(z) + iIm(z) and c = Re(c) + iIm(c), to yield Re(zn+1 ) = Re(zn )2 Im(zn )2 + Re(c) Im(zn+1 ) = 2 Re(zn ) Im(zn ) + Im(c) The Mandelbrot Set is the set of points in the complex parameter space c for which the iterated map variable z remains nite for an innite number of iterations. In practice, the maximum number of iterations is taken to be nite and the set is described on a 2D numerical lattice for C centered onto the origin. The beautiful images of the set that you might have already seen are produced based on the number of iteratons of the map that lead to an escape or that reach the maximum Google allowed. will give you thousands of hits on the Mandelbrot Set. Visit some.
Part A
Write a C (C++) code to generate the Mandelbrot Set Set a 2D grid in c. Default the range of c to be Re(c) = [-1. 0.8], Im(c) = [-1.0,1.0] and Nreal = 200, Nimg = 180. For each c values on the c grid iterate the map in z starting at z = (Re(z), Im(z)) = (0, 0) for a depth (maximum number) of iterations of 80 or until z escapes to innity The z escape criteria is based on the fact that z will necessarily escape if it reaches the region outside of a circle of radius 2 centered onto the origin z = 0 Output the number of iterations for each value of c in stdout 1
Part B
Produce a graph o...