Tiling a Sphere

thingiverse

This example utilizes the Pythagorean tiling on a sphere's surface, as explained by William E. Wenger in his web page. A cube is used for illustration, with a 2D pattern featuring 4-fold rotation axes. Nine copies of this pattern, extending infinitely, are shown in the image. In 3D, each cell contains 4-fold axes at its center and corners, with corner axes transforming to 3-fold ones. The tiling process requires six cube copies, visualized as images. Conversion to a sphere maintains 3-fold axes from the cube. OpenSCAD offers correct symmetry for Pythagorean solids' tiling, with various examples provided. Cube_with_pattern.stl and sphere_with_pattern_70.stl can be directly printed on build surfaces, while sphere_with_pattern_50.stl needs external support. The code for generating 2D dxf files and 3D stl files is also included in the p4_patterns.zip package. A printable 2D cube pdf file (p4_pattern_6.pdf) is available, which can be cut out and folded into a paper cube model.