Newton's Basin
From Math Images
Line 1: | Line 1: | ||
{{Image Description | {{Image Description | ||
|ImageName=Newton's Basin | |ImageName=Newton's Basin | ||
- | |Image= | + | |Image=NewtonBasin2.jpg |
|ImageIntro=Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function. | |ImageIntro=Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function. | ||
|ImageDescElem= | |ImageDescElem= | ||
[[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. | ||
Line 12: | Line 11: | ||
- | The animation emphasizes the roots in a Newton's Basin, whose equation clearly has three roots. The image | + | 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. |
|ImageDesc=The featured image on this page is a visual representation of Newton's Method for calculus expanded into the [[Complex Numbers|complex plane]]. To read a brief explanation on this method, read the following section entitled ''Newton's Method''. | |ImageDesc=The featured image on this page is a visual representation of Newton's Method for calculus expanded into the [[Complex Numbers|complex plane]]. To read a brief explanation on this method, read the following section entitled ''Newton's Method''. | ||
Line 53: | Line 52: | ||
[[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, as well as the featured image, 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. |
[[Image:NewtonFractalZoom.png|600px|center|Newton Basin with 3 Roots]] | [[Image:NewtonFractalZoom.png|600px|center|Newton Basin with 3 Roots]] | ||
- | |||
- | |||
Line 68: | Line 65: | ||
|other=Calculus | |other=Calculus | ||
- | |AuthorName= | + | |AuthorName=Ashley T. |
- | |AuthorDesc= | + | |AuthorDesc= |
- | |SiteName= | + | |SiteName=Fractal Foundation |
- | |SiteURL=http:// | + | |SiteURL=http://www.fractalfoundation.org/images/photo/3261826823/ashley-t-washington-ms.html |
|Field=Fractals | |Field=Fractals | ||
|Field2=Calculus | |Field2=Calculus |
Revision as of 11:56, 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 |
---|
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
- There are currently no teaching materials for this page. Add teaching materials.
Related Links
Additional Resources
- http://www.chiark.greenend.org.uk/~sgtatham/newton/ for further mathematical explanation
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.