# Earth Cubed

## The Coordinate System

Given that the surface of the eath is curved one might think that it would be difficult to construct a coordinate system that behaves like a rectangular coordinate system on a small scale but curves around the earth on a large scale.

To approach this problem we ask. How is a sphere like a cube? To a topologist they are nearly identical as it doesn’t take much effort to deform one into the other.

Imagine we had a cubic balloon. We blow it up a little and it is a cube. We draw a separate grid on each face of the cube shaped balloon. Then we blow it up more until it becomes a sphere. If our grid was fine enough then even with the balloon inflated to a spherical shape it still might look like a grid of squares is covering the balloon.

This should give confidence that we can divide the surface of a sphere up into sections that look like squares even though they cover a round object. So to do this we start thinking about what a square balloon might look like if it was inflated into a sphere.  The center of the top face will be mapped to the north pole. The center of the bottom face will be mapped to the south pole, the four faces on the side of the cube will divide the earth into four more sections. And the center point of each of these sections will be mapped to somewhere on the equator.

There are 360 degrees of longitude in the earth. Thus each section covers 90 degrees of longitude. If  we take the prime meridian as one edge then each face adjacent to the prime merridian will be centered at +/-45 degrees longitude.  Additionally the center points of the remain faces will be at +/-135 degrees of longitude.

These results are summarized in the following table. The remaining columns will be discussed in another post:

 gridPoints id parent level left right top bottom up down lat long time P_Theory 1 0 1 5 4 6 3 0 0 90 0 12 0.5 2 0 1 5 4 3 6 0 0 -90 0 12 0.5 3 0 1 5 4 1 2 0 0 0 45 12 0.5 4 0 1 3 6 1 2 0 0 0 135 12 0.5 5 0 1 6 3 1 2 0 0 0 -45 12 0.5 6 0 1 4 5 2 1 0 0 0 -135 12 0.5