Newton's Method in calculus is a procedure to find roots of polynomials, using an estimated value as a starting point.

Newton devised an iterated method (animated to the right) with the following steps:

1. Estimate a starting value () on the graph near to the root
2. Find the tangent line at that starting value
3. Find the root of the tangent line
4. Using the tangent's root as new starting value (), iterate the method to find a better estimate

The results of this method lead to very close estimates to the root of the polynomial. Newton's Method can also be expressed algebraically as follows, where is the nth estimate:

Creating Newton's Basin

Newton Basin with 5 Roots

To produce an interesting fractal, the Newton Method needs to be extended to the complex plane and to imaginary numbers. Newton's Basin is created using a complex polynomialOr a polynomial with co-efficients that are complex, such as , where z is in the form a + bi, with real and/or complex roots. In addition, each root in a Newton's Basin fractal is usually given a distinctive color. Thus, the fractal on the left has a total of five roots colored magenta, yellow, red, green, and blue.

Every pixel in the image is assigned a complex number coordinate. The coordinates are applied to the equation and iterated continually with the output of the previous iteration becoming the input of the next iteration. If the iterations lead the x-values of the coordinates to converge towards a particular root, the pixel is colored accordingly. If the iterations lead to a loop and not a root, then the pixel is usually colored black because the x-values do not converge.

Each root has a set of initial (or pixel) coordinates that converge to the root. This set of coordinates that are complex number values is called the root's basin of attraction- where the name of this fractal comes from. In addition, some images including shading in each basin. The shading is determined by the number of iterations (aka. escape time) it takes each pixel to converge to a particular root, and it allows us to see the location of the root more clearly.

Solutions

For example, the image below, as well as the featured image, was created from the equation . Since this equation is a 3rd degree complex polynomial, it has three roots, two of which are complex: z = -1.7693, 0.8846 + 0.5897i, and 0.8846 - 0.5897i. The resulting map of these solutions are to the right, and you can see that the Newton's Basin created from this complex polynomial has three roots (yellow, blue, and green) that correspond to the solution map.

Self-Similarity

As with all other fractals, Newton's Basin exhibits self-similarity. The video below is an interactive representation of the continual self-similarity displayed by a Newton's Basin with a root degree of 5 (similar to the fractal shown in the previous section). Towards the end of the video, you will notice that the pixels are no longer adequate to continue magnifying the image...however, the fractal still goes on.