Hill-and-valley weaving of voxel object surfaces

Hill-and-valley weaving must follow the medial of a bipartite map. The skeletal surface graphs of voxelized objects (example above) are bipartite. Pasting the truchet tile (below) onto each square face (with corner colors matching) gives a face 3-coloring of the medial graph (the new edges trace the boundaries between colors).

For example, for a single voxel, i.e., a cube, decorating its six faces with the truchet tile, shows that its 3-colored medial is a cuboctahedron (below) with its square faces colored 'saddle' and its triangular faces colored alternately 'hill' and 'valley'.

A spherical cuboctahedron (see below; art by Watchduck) is four great circles in an arrangement of maximum symmetry, so the smallest angles between these planes is equal to the dihedral angles of the tetrahedron, or approximately 70.5288 degrees.

To weave the cuboctahedron weave 'flat' (i.e., without hills and valleys) weavers must cross each other such that the internal angles of the triangular faces are about 70.5288 degrees, a bit wider than the 60 degrees these angles would measure on the plane. To make the triangular faces into hills and valleys, and, correspondingly, the square faces into saddles, we need even wider internal angles in the triangular faces. Below is a hill-and-valley weaving of the cuboctahedron with 100 degree internal angles in the triangular faces. Because of the hills on alternate triangles of the cuboctahedron, the basket appears strongly tetrahedral.