Tiling a Sphere

Here is a simple illustration of the Pythogorean tiling of a sphere. The idea comes from a web page by Willian E. Wenger which explains how to carve a periodic tiling on the surface of a sphere.

http://www.miracerros.com/artwork/g_sphere_layout.htm

A cube was used for this example. A simple two dimensional pattern with 4 fold rotation axes was generated. Nine copies of the pattern, which extends to infinity, are shown in the image above. Each cell has a 4 fold rotation axis at the center and a 4 fold rotation axis at each corner. One interesting thing is that the 4 fold axes at the corners are converted to 3 fold axes as the pattern goes from 2 to 3 dimensions.


To tile the sphere six copies are transferred to a cube - see images. The conversion to three dimensions changes the 4 fold axes at the corners to 3 fold axes on the cube. The 3 fold axes are preserved when the cube is converted to a sphere.


Three dimensional objects are the cube with the pattern transfer