In two dimensions, the regular square tiling has the awkwardness that there exist pairs of squares which touch just at a corner. It makes the adjacency graph non-planar. The regular hexagon tiling does not have this problem; any pair of hexagons either shares an edge or does not touch at all.
In three dimensions, polyhedra may touch at a face, an edge, or a vertex. Is there a tiling polyhedron that if two polyhedra touch, they share positive surface area? Or share at least a face or an edge (not touching just at a vertex)?
I suspect not. In two dimensions, the dual of the hexagon tiling (the adjacency graph) consists of simplices (triangles). In three dimensions, tetrahedra do not tile.
See the Wikipedia article on the bitruncated cubic honeycomb.
ReplyDeleteAlthough, if you want to generalize the adjacency graph of the hexagonal tiling to 3d, I think the diamond lattice (which is not the graph of a nice polyhedral tiling) may fit better: see my paper arxiv:0807.2218