# Harmonic Warping

(Difference between revisions)
 Revision as of 16:07, 23 June 2009 (edit)← Previous diff Revision as of 16:16, 23 June 2009 (edit) (undo)Next diff → Line 13: Line 13: |ImageDesc= |ImageDesc= [[Image:HarmonicWarp.png|thumb|300px]] [[Image:HarmonicWarp.png|thumb|300px]] + [[Image:UnionFlag.gif|300px]] + [[Image:UnionFlag_Rectangular.jpg|300px]] Essentially, an equation was used to map the points of Essentially, an equation was used to map the points of Line 30: Line 32: ==Four Infinite Poles== ==Four Infinite Poles== + [[Image:UnionFlag_4Poles.jpg|300px]] Link to [[Hyperbolic Geometry]] Link to [[Hyperbolic Geometry]] Line 36: Line 39:
Name Saint Andrew's Flag Saint Andrew's Flag Saint George's Flag Saint George's Flag
FlagOriginal Flag[[Image:StAndrews_Flag.png]][[Image:StAndrews_Flag.png|200px]][[Image:StGeorges_Flag.png]][[Image:StGeorges_Flag.png|200px]]
Rectangular Tiling Rectangular Tiling [[Image:StAndrews_Rectangular.jpg]] [[Image:StAndrews_Rectangular.jpg|200px]] [[Image:StGeorges_Rectangular.jpg]] [[Image:StGeorges_Rectangular.jpg|200px]]
Polar Tiling Polar Tiling [[Image:StAndrews_Polar.jpg]] [[Image:StAndrews_Polar.jpg|200px]] [[Image:StGeorges_Polar.jpg]] [[Image:StGeorges_Polar.jpg|200px]]
Four Infinite Poles [[Image:StAndrews_4Polar.jpg]] [[Image:StAndrews_4Polar.jpg|200px]] [[Image:StGeorges_4Polar.jpg]] [[Image:StGeorges_4Polar.jpg|200px]]
- + - + - + - + - + - + - + - + - Four Infinite Poles - + - +

## Revision as of 16:16, 23 June 2009

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

## Four Infinite Poles

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

# 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.