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.

Cube_with_pattern.stl and sphere_with_pattern_70.stl can be printed directly on the build surface. Sphere_with_pattern_50.stl needs external support.

The openSCAD code that generates the two dimensional dxf files is included as is the code that makes the three dimensional stl files. Dxf files and pdf files are in p4_patterns.zip.

A pdf file that shows the cube in 2 dimensions is included - p4_pattern_6.pdf. It can be printed, cut out, and folded to form a paper version of the cube.