#### Individual Presentation or Panel Title

Newton’s Fractals All Around: Newton’s Method and Fractals

#### Abstract

Newton’s method is a topic that is accessible to calculus students. We use Newton’s method to approximate the zeros of a given function using an iterative process. When considering a function with several zeros, the initial point determines which root of the function we obtain. However, this method can be extended to the complex plane, where we use Newton’s method to produce several interesting fractals. This project investigates Newton’s function, N(x) = x - f(x)/f’(x) and shows how to determine decimal approximations to the roots of f(x) for both real and complex valued functions. We also examine the basins of attraction for roots of both real and complex valued functions. Our work gives several examples to show how Newton’s method and the corresponding basins of attraction create amazing designs in the complex plane for a given function. It is interesting to see how a simple idea can be extended to new settings and reveal complicated and beautiful fractal patterns.

#### Location

Glass Dining Room

#### Start Date

3-5-2014 3:30 PM

#### End Date

3-5-2014 4:20 PM

Newton’s Fractals All Around: Newton’s Method and Fractals

Glass Dining Room

Newton’s method is a topic that is accessible to calculus students. We use Newton’s method to approximate the zeros of a given function using an iterative process. When considering a function with several zeros, the initial point determines which root of the function we obtain. However, this method can be extended to the complex plane, where we use Newton’s method to produce several interesting fractals. This project investigates Newton’s function, N(x) = x - f(x)/f’(x) and shows how to determine decimal approximations to the roots of f(x) for both real and complex valued functions. We also examine the basins of attraction for roots of both real and complex valued functions. Our work gives several examples to show how Newton’s method and the corresponding basins of attraction create amazing designs in the complex plane for a given function. It is interesting to see how a simple idea can be extended to new settings and reveal complicated and beautiful fractal patterns.