Newton's Basin

Newton's Basin
Fields: Fractals and Calculus
Image Created By: Ashley T.
Website: Fractal Foundation

Newton's Basin

Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function.

Basic Description

Animation Emphasizing Roots

This image is one of many examples of Newton's Basin or Newton's Fractal. Newton's Basin is based on a calculus technique called Newton's Method, a procedure Newton developed to estimate rootsA root is the x-value where y = 0 and the graph of an equation crosses the horizontal x-axis (or solutions) of equations.

Each pixel in a Newton's Basin corresponds to a unique coordinate, or point. The colors in a Newton's Basin usually correspond to each individual root of the equation, and can be used to infer where each root is located. Each color region reflects the set of points, which, after undergoing iteration with the equation describing the fractal, will eventually get closer and closer to the value of the root associated with that color.

The animation emphasizes the roots in a Newton's Basin, whose equation clearly has three roots. The image featured at the top of this page is also a Newton's Basin with three roots.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Calculus

[Click to view A More Mathematical Explanation]

The image at the top of this page is a visual representation of Newton's Method in calculus expanded [...]

[Click to hide A More Mathematical Explanation]

The image at the top of this page is a visual representation of Newton's Method in calculus expanded into the complex plane.

Newton's Method

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 x-value ($x_o$) on the graph near to the root
2. Find the tangent line at that starting x-value
3. Find the root of the tangent line
4. Using the tangent's root as new starting x-value ($x_{n}, x_{n+1},...$), 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 $x_n$ is the nth estimate:

$f'(x_n) = \frac{\mathrm{\Delta y}}{\mathrm{\Delta x}} = \frac{0 - y_n}{x_n - x_{n+1}}$

$f'(x_n) = \frac{f(x_n)}{x_n - x_{n+1}}$

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Newton's Basin

Newton Basin with 5 Roots

To produce an interesting fractal, the Newton Method needs to be extended to the complex plane. Newton's Basin is created using a complex polynomialOr a polynomial with co-efficients that are complex, such as $p(z) = z^3 - 2z + 2$, 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 right is generated by a polynomial with a total of five roots colored magenta, yellow, red, green, and blue.

Every pixel in the image represents a complex number. Each complex number is applied to the equation and iterated continually with the output of the previous iteration becoming the input of the next iteration. This iteration is done by using the same equations discussed in the previous Newton's method section, where x is now a complex number z, y is now a complex number p, and $z_n$ is the nth estimate:

$f'(z_n) = \frac{\mathrm{\Delta p}}{\mathrm{\Delta z}} = \frac{f(z_n)}{z_n - z_{n+1}}$

$z_{n+1} = z_n - \frac{f(z_n)}{f'(z_n)}$

Coloring

$f(z) = z^5 - 1$

If the iterations lead the complex number to converge towards a particular root, the pixel is colored according to the color of that root. If the iterations lead to a loop and not a root, then the pixel is colored black because the complex number does not converge.

Each root has a set of complex numbers (or pixels)that converge to the root (algebraically, this set would include all of the $z_0$ values referenced above). This set of coordinates 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 it takes each pixel to converge to its root, and it allows us to see the location of the root more clearly. The darker the shading of a pixel is, the more iterations it requires for that pixel to converge to its respective root.

An Example

For example, the image below, as well as the image at the top of the page, was created from the equation $p(z) = z^3 - 2z + 2$. Since this equation is a 3rd degree complex polynomial, it has three roots, two of which are complex:

$z_1 = -1.7693$

$z_2 = 0.8846 + 0.5897i$

$z_3 = 0.8846 - 0.5897i$

The resulting map of these solutions are to the right. 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 the Newton's Basin shown in an above section with a root degree of 5 $f(z) = z^5 - 1$. 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.

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Teaching Materials

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References

Wikipedia, Newton fractal page and Newton's Method page

Simon Tatham, Fractals derived from Newton-Raphson iteration

David E. Joyce, Newton Basins

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