Newton's Basin
From Math Images
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- | + | [[Image:NewtonFractalZoom.png|500px|thumb|right|Close up of Newton Basin with 3 Roots]] | |
For example, the image above was created from the equation <math> p(z) = z^3 - 2z + 2</math>. | For example, the image above was created from the equation <math> p(z) = z^3 - 2z + 2</math>. | ||
roots... | roots... | ||
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+ | [[Image:Roots.gif]] | ||
====Self-Similarity==== | ====Self-Similarity==== | ||
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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. | 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. | ||
{{#ev:tubechop|gh6e95OmoAk&start=5&end=70|300|left}} | {{#ev:tubechop|gh6e95OmoAk&start=5&end=70|300|left}} |
Revision as of 11:31, 4 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
- 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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