Newton's Basin
From Math Images
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====Creating Newton's Basin==== | ====Creating Newton's Basin==== | ||
- | + | [[Image:NewtonBasin_5Roots.gif|thumb|left|200px|Newton Basin with 5 Roots]] | |
- | [[Image:NewtonBasin_5Roots.gif|thumb|200px|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 <balloon title="load:Content">complex polynomial</balloon><span id="Content" style="display:none">Or a polynomial with co-efficients that are complex, such as <math> p(z) = z^3 - 2z + 2</math></span>, 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. |
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- | 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 <balloon title="load:Content">complex polynomial</balloon><span id="Content" style="display:none">Or a polynomial with co-efficients that are complex, such as <math> p(z) = z^3 - 2z + 2</math></span>, with real and/or complex roots. In addition, in a Newton's Basin fractal | + | |
Each 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 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. | ||
+ | Therefore, each root has a set of initial (or pixel) coordinates <math>x_0</math> 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. | ||
- | + | <gallery caption="" widths="100px" heights="100px" perrow="2"> | |
- | + | Image:NewtonFractal_Zoom.png|Newton Basin with 3 Roots | |
- | + | Image:NewtonFractal_ZoomClose.png|Close up of Newton Basin with 3 Roots | |
- | + | </gallery> | |
- | + | For example, the image above was created from the equation <math> p(z) = z^3 - 2z + 2</math>. | |
- | + | roots... | |
+ | wolfram alpha image | ||
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====Self-Similarity==== | ====Self-Similarity==== |
Revision as of 14:55, 3 June 2009
- 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 |
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Contents |
Basic Description
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 Basin
Teaching Materials
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About the Creator of this Image
Nicholas Buroojy has created many math images including Newton's Lab fractals, Julia and Mandelbrot Sets, Cantor Sets...
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