Newton's Basin
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[[Image:NewtonBasin_Animate.gif|thumb|left|200px|Animation Emphasizing Roots]] | [[Image:NewtonBasin_Animate.gif|thumb|left|200px|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 concept called Newton's Method, a procedure Newton developed to estimate a <balloon title="load:myContent">root</balloon><span id="myContent" style="display:none">A root is located where y = 0 and the graph of an equation crosses the horizontal x-axis [[Image:Root.gif|200px]]</span> of an equation. | 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 <balloon title="load:myContent">root</balloon><span id="myContent" style="display:none">A root is located where y = 0 and the graph of an equation crosses the horizontal x-axis [[Image:Root.gif|200px]]</span> of an equation. | ||
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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 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. | ||
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The animation emphasizes the roots in a Newton's Basin, whose equation clearly has three roots. The image featured on this page is also a Newton's Basin with three roots, presented more artistically. | The animation emphasizes the roots in a Newton's Basin, whose equation clearly has three roots. The image featured on this page is also a Newton's Basin with three roots, presented more artistically. |
Revision as of 10:56, 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 featured on this page 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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Related Links
Additional Resources
- http://www.chiark.greenend.org.uk/~sgtatham/newton/ for further mathematical explanation
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