Wednesday, October 4, 2017

Isohedral classification

I have finally begun to understand the main thrust of Grünbaum and Shepard's classification of isohedral tilings, though there are details that still elude me. They begin with the eleven Laves tilings, tilings in which the angles at each vertex are equal. In the tiling below there are vertices with three line converging, so all the angles at these vertices are 120 degrees. There are also vertices with six lines converging, so the angles here must be 60 degrees.

Grünbaum and Shepard then consider the various rotational and reflectional symmetries that each of the tiles can have. There are ten for the regular hexagon, with twofold, threefold, and sixfold rotational symmetry, plus reflection over one side, three sides, one diagonal, three diagonals, reflection over one diagonal and one edge, reflection over three sides and three diagonals, and no symmetry at all. They then see how each of these possibilities can fit together consistently. Each of those ten yield at least one isohedral class; the one with no symmetry yields the seven hexagonal Heesch types.

Can there be a class formed from reflectional symmetry in the above figure? The answer is, "No". If we bisect the tile, we get the result on the bottom in the figure. Notice that some of the tiles have five neighbors and some have only four. The symmetry line cuts the base of the tile but the base of the tile never abuts another base. It would need to do so for it to make fit into its own isohedral class.  The only isohedral class that the tiling above yields is IH21, which is Heesch type CC3C3C6C6.