Students are encouraged to complete the assignment prior to the recitation. This assignment will not be collected nor graded.
The Mandelbrot set is a set of points C on the complex plane for which the value of
= "& + , + = 0 , remains bounded for all n. The above image shows the set plotted on
the complex plane (x is the real axis, y is imaginary, and black indicates points in the set).
The boundary of this set is a two-dimensional fractal. You can interactively view the Mandelbrot set at math.hws.edu/eck/js/mandelbrot/MB.html . Try repeatedly zooming in on the boundaries of the set and see what you find!
For any complex number C, it is known that once the absolute value of " becomes greater than 2, the point C is not in the Mandelbrot set. To approximate this set, one chooses points on a grid and iterates the equation for each of these points until reaching a predefined maximum n or until the point is determined not to be in the set. For plotting, each point or grid cell is colored based on the maximum n value for which | "| < 2.
1. Plot the Mandelbrot set for a 4x4 grid centered around the origin (x is the real axis, y is the imaginary axis) with a cell size of 0.1x0.1 and max n=10
2. Plot the Mandelbrot set for a 2x2 grid centered around the origin (x is the real axis, y is the imaginary axis) with a cell size of 0.01x0.01 and max n=10
3. Plot the Mandelbrot set for a 2x2 grid centered around the origin (x is the real axis, y is the imaginary axis) with a cell size of 0.01x0.01 and max n=100
4. Plot the Mandelbrot set for a 2x2 grid centered around the origin (x is the real axis, y is the imaginary axis) with a cell size of 0.001x0.001 and max n=100
