How to Carve a Periodic Tiling on the Surface of a Sphere

    1. Select or create a 2-dimensional tiling with 3, 4, or 5-fold rotational symmetry.

    A tiling with 3-fold symmetry (illustrated above) will fit onto the equilateral triangular faces of a tetrahedron, an octahedron or an icosahedron.   A tiling with 4-fold symmetry will fit onto the square faces of a cube,   and a tiling with 5-fold symmetry will fit onto the pentagonal faces of a dodecahedron.   These five regular polyhedrons are known as the platonic solids.   They can each be perfectly circumscribed by a sphere.

    2. Divide the 2-dimensional tiling into triangle, square, or pentagon-shaped "faces" to match those on the surface of the platonic solid.

    This division is done by connecting the points of rotation on the 2-dimensional tiling with straight lines.   The lines should form a grid of equilateral triangles,   squares,   or pentagons.   You can confirm that your 2-dimensional tessellated drawing will tile the solid surface,   and get a preview of your finished spherical carving,   by making cut-outs like the ones above,   which can be folded up to form the platonic solid figure.

    3. Inscribe the faces of the platonic solid on the surface of the sphere.

In the drawings above, the lines on the spheres are great circles that intersect at angles of 60 degrees (tetrahedron),   45 degrees (cube and octahedron),   or 36 degrees (dodecahedron and icosahedron).

4. Transfer your 2-dimensional drawing to the surface of the sphere

When you transfer your drawing to the triangular,   square,   or pentagonal faces you have outlined on the surface of the sphere (shown shaded in the drawings above) you will have to distort them a bit to fit the curved surface.   But by working one-face-at-a-time,   that becomes manageable.

5. Carve the sphere