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

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.

[Show Limit Proof][Hide Limit Proof]

The graph to the right shows clearly that d(x) approaches 1 as x goes to infinity. Mathematically:

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:

Union Flag (source image)

Union Flag tiled infinitely into a unit square

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

Link to Hyperbolic Geometry

Comparing the Different Types of Tilings

	Saint Andrew's Flag	Saint George's Flag
Original Flag
Rectangular Tiling
Polar Tiling
Four Infinite Poles Tiling