Harmonic Warping
From Math Images
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|ImageName=Harmonic Warping of Blue Wash | |ImageName=Harmonic Warping of Blue Wash | ||
|Image=Harmonic warp.jpg | |Image=Harmonic warp.jpg | ||
- | |ImageIntro=This image is a tiling based on harmonic warping operations. These operations take a source image and compress it to show the infinite tiling of the source image | + | |ImageIntro=This image is a tiling based on harmonic warping operations. These operations take a source image and compress it to show the infinite tiling of the source image within a finite space. |
|ImageDescElem= | |ImageDescElem= | ||
- | This image is an infinite tiling. If you look closely at the edges of the image, you can see that the tiles become smaller and smaller and seem to fade into the edges | + | This image is an infinite tiling. If you look closely at the edges of the image, you can see that the tiles become smaller and smaller and seem to fade into the edges. The border of the image is infinite so that the tiling continues unendingly and the tiles become eternally smaller. |
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|ImageDesc= | |ImageDesc= | ||
- | + | To create this image, a harmonic warping operation was used to map the infinite tiling of the source image onto a finite plane. This operation essentially took the entire infinite <balloon title="Euclidean refers to the traditional geometric space that most people are initially exposed to, as opposed to non-Euclidean (which includes Hyperbolic and Elliptical geometry)"> Euclidean </balloon> plane and squashed it into a rectangular. This type of operation can be called a ''distance compressing warp''. | |
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+ | [[Image:HarmonicWarp.png|thumb|right|300px|Harmonic Warping Equation]] | ||
+ | The equation used to perform the harmonic warp is show in a graph to the right and is as follows: | ||
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+ | :<math>d(x) = 1 - \frac{1}{1+x}</math> | ||
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+ | :<math>d(y) = 1 - \frac{1}{1+y}</math> | ||
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+ | You can see that for both of these equations, as x and y go to infinity, d(x) and d(y) both approach a limit of 1. | ||
+ | {{HideThis|1=Limit|2= | ||
+ | <math>\lim_{x \rightarrow \infty} 1 - \frac{1}{1+x}</math> | ||
+ | <math>\lim_{x \rightarrow \infty} 1 - \frac{1}{1+\infty}</math> | ||
+ | <math>\lim_{x \rightarrow \infty} 1 - \frac{1}{\infty}</math> | ||
+ | <math>\lim_{x \rightarrow \infty} 1 - 0}</math> | ||
+ | <math>\lim_{x \rightarrow \infty} 1 </math> | ||
+ | }} | ||
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+ | [[Image:UnionFlag.gif|300px]] | ||
+ | [[Image:UnionFlag_Rectangular.jpg|300px]] | ||
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*mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle | *mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
- | <td>Four Infinite Poles</td> | + | <td>Four Infinite Poles Tiling</td> |
<td> [[Image:StAndrews_4Polar.jpg|200px]]</td> | <td> [[Image:StAndrews_4Polar.jpg|200px]]</td> | ||
<td> [[Image:StGeorges_4Polar.jpg|200px]]</td> | <td> [[Image:StGeorges_4Polar.jpg|200px]]</td> |
Revision as of 10:29, 24 June 2009
- This image is a tiling based on harmonic warping operations. These operations take a source image and compress it to show the infinite tiling of the source image within a finite space.
Harmonic Warping of Blue Wash |
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Contents |
Basic Description
This image is an infinite tiling. If you look closely at the edges of the image, you can see that the tiles become smaller and smaller and seem to fade into the edges. The border of the image is infinite so that the tiling continues unendingly and the tiles become eternally smaller.
The source image used for this tiling is another image that is mathematically interesting and is also featured on this website. See Blue Wash for more information about how the source image was created.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Single Variable Calculus
To create this image, a harmonic warping operation was used to map the infinite tiling of the source [...]
To create this image, a harmonic warping operation was used to map the infinite tiling of the source image onto a finite plane. This operation essentially took the entire infinite Euclidean plane and squashed it into a rectangular. This type of operation can be called a distance compressing warp.
The equation used to perform the harmonic warp is show in a graph to the right and is as follows:
You can see that for both of these equations, as x and y go to infinity, d(x) and d(y) both approach a limit of 1.
Failed to parse (syntax error): \lim_{x \rightarrow \infty} 1 - 0}
- mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle
Polar Harmonic Warping
Link to Polar Coordinates
Four Infinite Poles
Link to Hyperbolic Geometry
Big table of rectangular, polar, cardinal 4 poles for both flag!
Saint Andrew's Flag | Saint George's Flag | |
Original Flag | ||
Rectangular Tiling | ||
Polar Tiling | ||
Four Infinite Poles Tiling |
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
About the Creator of this Image
Paul Cockshott is a computer scientist and a reader at the University of Glasgow. The various math images featured on this page were originally produced for his research.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.