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 transferred to its faces:

cube_with_pattern.stl

and two versions of the cube converted to a sphere:

sphere_with_pattern_50.stl
sphere_with_pattern_70.stl

Look at the corners of the cube. Now consider what would happen if the cube were flexible and inflated to a sphere. On the sphere the centers of the cube faces are represented by 6 hypocycloids and the corners by 8 distorted triangles.

OpenSCAD creates the correct symmetry for the "tiling" of all the Pythogorean solids. See things 40276, 39977, 39818, and especially 39424 which has a similar example.