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.
1 comment :
See the Wikipedia article on the bitruncated cubic honeycomb.
Although, 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
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