Harmonic Warping

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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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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 (ex. Hyperbolic and Elliptical geometry)"> Euclidean </balloon> plane and squashed it into a square. 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 equations used to perform the harmonic warp is show in a graph to the right and is as follows:
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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>
:<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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:<math>d(y) = 1 - \frac{1}{1+y}</math>, where (x,y) is a coordinate on the Euclidean plane tiling and (d(x), d(y)) is a coordinate on the non-Euclidean square tiling
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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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You can observe for both of these equations that 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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{{HideThis|1=Prove Limit|2=
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The graph to the right shows clearly that d(x) approaches 1 as x goes to infinity. Mathematically:
<math>\lim_{x \rightarrow \infty}d(x) = 1 - \frac{1}{1+x}</math>
<math>\lim_{x \rightarrow \infty}d(x) = 1 - \frac{1}{1+x}</math>
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[[Image:UnionFlag.gif|400px]]
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Since d(x) and d(y) approach 1 as x and y go to infinity, the square plane that the infinite tiling is mapped to must be a unit square (that is its dimensions are 1 unit by 1 unit). Since the unit square fits an infinite tiling within its finite border, the square is not a traditional Euclidean plane. As the tiling approaches the border of the square, distance within the square increases non-linearly. In fact, the border of the square is infinite because the tiling goes on indefinitely.
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[[Image:UnionFlag_Rectangular.jpg|400px]]
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*mapping (x,y) from Euclidean plane unto (d(x),d(y)) in rectangle
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Here is another example of this type of tiling contained in a square using the Union Flag:
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[[Image:UnionFlag.gif|200px]]
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[[Image:UnionFlag_Rectangular.jpg|400px]]

Revision as of 10:47, 24 June 2009

Image:inprogress.png
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.

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

Harmonic Warping Equation
Harmonic Warping Equation
To create this image, a harmon [...]

Harmonic Warping Equation
Harmonic Warping Equation

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 square. This type of operation can be called a distance compressing warp.


The equations 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}, where (x,y) is a coordinate on the Euclidean plane tiling and (d(x), d(y)) is a coordinate on the non-Euclidean square tiling

You can observe for both of these equations that as x and y go to infinity, d(x) and d(y) both approach a limit of 1.

The graph to the right shows clearly that d(x) approaches 1 as x goes to infinity. Mathematically: \lim_{x \rightarrow \infty}d(x) = 1 - \frac{1}{1+x}

\lim_{x \rightarrow \infty}d(x) = 1 - \frac{1}{1+\infty}

\lim_{x \rightarrow \infty}d(x) = 1 - \frac{1}{\infty}

\lim_{x \rightarrow \infty}d(x) = 1 - 0

\lim_{x \rightarrow \infty}d(x) = 1


Since d(x) and d(y) approach 1 as x and y go to infinity, the square plane that the infinite tiling is mapped to must be a unit square (that is its dimensions are 1 unit by 1 unit). Since the unit square fits an infinite tiling within its finite border, the square is not a traditional Euclidean plane. As the tiling approaches the border of the square, distance within the square increases non-linearly. In fact, the border of the square is infinite because the tiling goes on indefinitely.


Here is another example of this type of tiling contained in a square using the Union Flag:


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

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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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