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. |