Newton's Basin

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[[Image:Roots.gif|thumb|right|Solutions <math> p(z) = z^3 - 2z + 2</math>]]
[[Image:Roots.gif|thumb|right|Solutions <math> p(z) = z^3 - 2z + 2</math>]]
For example, the image below was created from the equation <math> p(z) = z^3 - 2z + 2</math>. 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 solution 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.
For example, the image below was created from the equation <math> p(z) = z^3 - 2z + 2</math>. 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 solution 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.
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[[Image:NewtonFractalZoom.png|500px|thumb|center|Newton Basin with 3 Roots]]
+
[[Image:NewtonFractalZoom.png|600px|center|Newton Basin with 3 Roots]]

Revision as of 11:53, 4 June 2009

Image:inprogress.png
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.
Newton's Basin
Fields: Fractals and Calculus
Created By: Nicholas Buroojy

Contents

Basic Description

Animation Emphasizing Roots
Animation Emphasizing Roots
Newton Basin with 3 Roots
Newton Basin with 3 Roots

This image is one of many examples of Newton's Basin or Newton's Fractal. Newton's Basin is based on a calculus concept called Newton's Method, a procedure Newton developed to estimate a root of an equation.


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. The region of each color reflects the set of coordinates (x,y) whose x-values, after undergoing iteration with the equation describing the fractal, will eventually get closer and closer to the value of the root.


The animation emphasizes the roots in a Newton's Basin, whose equation clearly has three roots. The image to the right is also a Newton's Basin with three roots, presented more artistically.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Calculus

The featured image on this page is a visual representation of Newton's Method for calculus expanded i [...]

The featured image on this page is a visual representation of Newton's Method for calculus expanded into the complex plane. To read a brief explanation on this method, read the following section entitled Newton's Method.

Newton's Method

Newton's Method for calculus is a procedure to find a root of a polynomial, using an estimated coordinate as a starting point. Usually, the roots of a linear equation: y = mx + b can be simply found by setting y = 0 and solving for x. However, with higher degree polynomials, this method can be much more complicated.


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

  • Estimate a starting coordinate on the graph near to the root
  • Find the tangent line at that starting coordinate
  • Find the root of the tangent line
  • Using the root as the x-coordinate of the new starting coordinate, iterate the method to find a better estimate

The results of this method lead to very close estimates to the actual root. Newton's Method can also be expressed:

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


Newton's Basin

Creating Newton's Basin

Newton Basin with 5 Roots
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 polynomial, with real and/or complex roots. In addition, each root in a Newton's Basin fractal is usually given a distinctive color. It is clear that 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 x_0 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 it takes each pixel to converge to a particular root, and it allows us to see the location of the root more clearly.



Solutions
Solutions p(z) = z^3 - 2z + 2

For example, the image below 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 = -1.7693, 0.8846 + 0.5897i, and 0.8846 - 0.5897i. The resulting solution 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.

Newton Basin with 3 Roots




Self-Similarity

As with all other fractals, Newton's Basin exhibits self-similarity. The video to the left 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.




Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

About the Creator of this Image

Nicholas Buroojy has created many math images including Newton's Lab fractals, Julia and Mandelbrot Sets, Cantor Sets...


Related Links

Additional Resources

http://www.chiark.greenend.org.uk/~sgtatham/newton/ for further mathematical explanation





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