Harmonic Warping

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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 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
To create this image, a harmon [...]

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, 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 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

A tiling similar to the one mentioned above can be performed in polar coordinates. In this tilings

Link to Polar Coordinates

Four Infinite Poles

UnionFlag 4Poles.jpg Link to Hyperbolic Geometry

Comparing the Different Types of Tilings

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

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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 emerged from his research dealing with digital image processing.

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