COURSES Alejandro B. Engel Department of Mathematics and Statistics Rochester Institute of Technology email: abesma@rit.edu Phone: (716) 475 2123 NICE MATH. RESEARCH OTHER

 FRACTALS Lines, curves and circles are one dimensional geometric objects. Rectangles, triangles and disks are two dimensional geometric objects. Is there something "in between" one dimensional geometric objects and two dimensional geometric objects? Intuitively, following our common sense one would tend to answer NO. However, that what separates the black from the colored in the picture above is "more" than a curve, but "less" than a rectangle: it is a FRACTAL. The name fractal comes from the idea that its dimension is a number larger than one and smaller than two. Fractional dimension is what fractal brings to mind. The most striking feature of a fractal is self similarity. Any section of a fractal will reproduce the fractal when magnified enough. Magnification will just keep reproducing the fractal stucture, without end. Experiment with the applet below to experience self similarity. If you want to learn a little about the concept of fractals and their associated Mandelbrot and Julia Sets, follow the sequence of screens hyperlinked at your left. Do not skip any screen, read all the text and study carefully the illustrations: play with them as much as possible. Take the Mandelbrot Map, F(z) = z2 + c; with both z and c complex numbers. This function maps the complex plane into itself. Now start with any fixed number c, and let: z0 = 0, z1=F(z0), z2=F(z1), ... ,zn+1=F(zn), and so on....These are called the ITERATES of F(z). There are only two possibilities: either the iterates of F(z) remain bounded -that is within a disk center at the origin-, or they do not remain bounded -they will eventually leave any disk centered at the origin-. In the complex plane that contains c, for every point do the following: use the value of this point in function F(z), if the iterates of F(z) remain bounded paint the point green, otherwise paint it yellow. That what separates green from yellow is a fractal. It is the Mandelbrot Set.