Anne Burns
Visualization is an important aid in the study of mathematics. Often a diagram or graph can be understood more easily than pages of computations. In many of my courses, particularly those in math education, I have the students buy an inexpensive set of colored markers and use them in a variety of topics from abstract algebra to complex roots of a function.
Each of the disks in the 3X3 matrix of disks is a picture of the first five backward iterations of f(z)=z^n+c/z^m where c is a small positive real number. The rows represent n=2,3,4 and the columns represent m=2,3,4. The black disks in the center consist of the set of points z such that |f(z)|>1.1. The second largest sets of disks are blue; they are the inverse images of the black disks under f; ochre disks are the inverse images of blue disks; red disks are the inverse images of ochre disks, etc. First notice the n+m symmetry in each disk. Next, can you identify n and m by this pattern? Hint: choose one blue disk in each entry and count the number of pre-images closer to the center and the number of pre-images further away from the center.