# The Mandelbrot Set

## Department of Mathematics

### The Mandelbrot Set

(summary of the Original Written Report)

Date: April 2004

Introduction

Consider the sequence given by

where c = a + bι. The Mandelbrot Set is the set of points (a,b) such that the sequence {Zi} is bounded. Adam's program, "The Mandelbrot Set" (linked below), allows you to explore Mandelbrot sets for different values of n, and calculates the area of the given set. It also allows you to take a closer look at the set by zooming in on sections as you desire, and recalculates the area of the displayed set.

The Julia Set, another iterative, complex mapping, is also explored in this program. Under the "Julia Set Applications" menu option, you can regenerate the Mandelbrot Set for your chosen value of n, and then click on any point in the Mandelbrot Set and the program will generate the corresponding Julia Set!

Objectives

Adam's objectives in this project were to write a program that would graphically display the Mandelbrot set, and the corresponding Julia Set.

Results

Adam found that initially, as the exponent n is increased, the area of the generated regions also increased.  However, eventually, at some value of n, the area starts to converge back to a fixed value. This led him to formulate his own mathematical conjecture which you can find in the program under the heading "Observations and Findings".