Difference between revisions of "Harmonic Warping"

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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 into a finite space.
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|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 is true. The border of the image is infinite so that the tiling is infinite and the tiles become infinitely smaller.
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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=
[[Image:HarmonicWarp.png|thumb|300px]]
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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]]
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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.
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{{HideThis|1=Limit|2=
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<math>\lim_{x \rightarrow \infty} 1 - \frac{1}{1+x}</math>
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<math>\lim_{x \rightarrow \infty} 1 - \frac{1}{1+\infty}</math>
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<math>\lim_{x \rightarrow \infty} 1 - \frac{1}{\infty}</math>
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<math>\lim_{x \rightarrow \infty} 1 - 0}</math>
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<math>\lim_{x \rightarrow \infty} 1 </math>
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}}
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[[Image:UnionFlag.gif|300px]]
 
[[Image:UnionFlag.gif|300px]]
 
[[Image:UnionFlag_Rectangular.jpg|300px]]
 
[[Image:UnionFlag_Rectangular.jpg|300px]]
Essentially, an equation was used to map the points of values
 
 
 
  
*equation <math>d(x) = 1 - \frac{1}{1+x}</math>, limit is 1
 
<math>d(y) = 1 - \frac{1}{1+y}</math>, limit is 1
 
  
* distance compressing warp
 
*infinite tiling of Euclidean plane mapped onto a rectangle (or ellipse)
 
 
*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>
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<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 09:29, 24 June 2009

Inprogress.png
Harmonic Warping of Blue Wash
Harmonic warp.jpg
Fields: Calculus and Fractals
Image Created By: Paul Cockshott
Website: Fractal Art

Harmonic Warping of Blue Wash

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.


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.


Harmonic Warping Equation

The equation used to perform the harmonic warp is show in a graph to the right and is as follows:

d(x) = 1 - \frac{1}{1+x}
d(y) = 1 - \frac{1}{1+y}

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.

\lim_{x \rightarrow \infty} 1 - \frac{1}{1+x} \lim_{x \rightarrow \infty} 1 - \frac{1}{1+\infty} \lim_{x \rightarrow \infty} 1 - \frac{1}{\infty} Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \lim_{x \rightarrow \infty} 1 - 0}

\lim_{x \rightarrow \infty} 1


UnionFlag.gif UnionFlag Rectangular.jpg


  • mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle


Polar Harmonic Warping

Link to Polar Coordinates


Four Infinite Poles

UnionFlag 4Poles.jpg 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 StAndrews Flag.png StGeorges Flag.png
Rectangular Tiling StAndrews Rectangular.jpg StGeorges Rectangular.jpg
Polar Tiling StAndrews Polar.jpg StGeorges Polar.jpg
Four Infinite Poles Tiling StAndrews 4Polar.jpg StGeorges 4Polar.jpg




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.








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Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.