# Harmonic Warping

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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 into 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. This is true. The border of the image is infinite so that the tiling is infinite and the tiles become infinitely 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

[[Image:UnionFlag_Rectangular.jp [...]

Essentially, an equation was used to map the points of

• equation $d(x) = 1 - \frac{1}{1+x}$, limit is 1

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

## 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

# 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.

If you are able, please consider adding to or editing this page!

Have questions about the image or the explanations on this page?
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