# Edit Edit an Image Page: Harmonic Warping

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 Image Title*: Upload a Math Image 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. 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 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. 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'''. [[Image:HarmonicWarp.png|right|thumb|302px|Harmonic Warping Equation]] The equations used to perform the harmonic warp is show in a graph to the right and is as follows, 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 :$d(x) = 1 - \frac{1}{1+x}$ :$d(y) = 1 - \frac{1}{1+y}$ 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|limit]] of 1. {{HideThis|1=Limit Proof|2= 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: Image:UnionFlag.gif Paul Cockshott is a computer scientist and a reader at the University of Glasgow. The various math images featured on this page emerged from his research dealing with digital image processing. Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other Paul Cockshott, [http://www.dcs.gla.ac.uk/~wpc/ Paul Cockshott] I suggest adding a section on why this operation is called ''Harmonic'' warping and expanding the '''Polar Tiling''' section. Yes, it is.