The GDM in Dimension Three

The Generalized Dual Method, or GDM, itself generalizes to higher Euclidean spaces. In 3-space, as mentioned in my last, an arrangement of planes, expressed in the form {x,y,z,d}, leads to a tessellation of zonohedra. And although we may give the GDM planes in arbitrary, "general" position, more typically we use a set of symmetry vectors in the 3-space.

For instance, we could pick the edge-centers of the Platonic Icosahedron; there are thirty edges, but they occur in oppposite pairs. This gives problems to the GDM, so we pick one out of each of the fifteen pairs, it doesn't matter which one, to obtain fifteen independent vectors.

Were we to make merely fifteen planes, and let them all pass through the origin, that is, each plane of the form {x,y,z,0}, we'd obtain a "degenerate" tiling consisting of one undivided zonohedron, the Archimedean Truncated Icosidodecahedron.

But, suppose we make thirty planes, fifteen at distance 0 as above, the other fifteen at distance 1. We obtain a Truncated Iocosidodecahedron of edge 2, composed of 176 separate zonohedra of edge 1; and, perforce, since fifteen of the planes do intersect at one point (those with distance 0), there is one Truncated Icosidodecahedron of edge 1 among the 176 zonohedra.

But it is not symmetrically situated within the tiling. It is entirely to one side, although, of course, entirely within the Truncated Icosidodecahedron of edge 2.

The animation above, made using the software, Mathematica, builds up the tiling zonohedron by zonohedron, beginning at the innermost and ending with the outermost.