Harmonic Warping

Harmonic Warping of Blue Wash
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 The source image refers to the actual image that is being tiled. In this case: 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

[Click to view A More Mathematical Explanation]

To create this image, a harmonic warping operation was used to map the infinite tiling of the s [...]

[Click to hide A More Mathematical Explanation]

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 Euclidean refers to the traditional geometric space that most people are initially exposed to, as opposed to non-Euclidean (ex. hyperbolic geometry and elliptic geometry) plane and squashed it into a square. This type of operation can be called a distance compressing warp.

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 of 1.

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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:

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  Union Flag (source image)

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  Union Flag tiled infinitely into a unit square

Polar Harmonic Warping

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A polar tiling

A tiling similar to the one mentioned above can be performed in polar coordinates. Polar tilings are infinite Euclidean tilings condensed into a finite circular space with an infinite border. The harmonic warping operations used to map the infinite Euclidean plane onto the circle are performed based on the radius of the circle being 1 unit.

Four Infinite Poles

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  Crosses continue to intersect at 90 degrees

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  "X"'s collapse to the four poles

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  Union Flag Four Infinite Pole Tiling

Another tiling that is possible involves designating the four cardinal poles of the circular border as infinite areas. The tiling then becomes very similar to a tiling done in the Poincaré Disk Model representing hyperbolic geometry. If the center of the circle of this type of tiling corresponds to the Euclidean origin where the x and y axes meet, then we can see that the tiling is consist with the source image. In the Poincaré Disk Model, straight lines are represented as circles and perpendicular lines still intersect at an angle of 90 degrees. A break down of the four infinite pole tiling of the Union Flag above shows that in each tile, the cross of the Union Flag still intersects with the x axes and the "X" of the Union Flag collapses towards each of the four poles.

Comparing the Different Types of Tilings

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	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 emerged from his research dealing with digital image processing.

References

Paul Cockshott, Paul Cockshott

Future Directions for this Page

I suggest adding a section on why this operation is called Harmonic warping and expanding the Polar Tiling section.

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