Figure 4: Format of the tiles appearing in the evolution of Rule
110, as it affects the appearance of the T3 tile.
Having agreed on the use of triangles as tiles, they can be labelled Tn, where n is the number of zeroes in their top row, and presented as being completely bordered, or semibordered, as in Table 4. The semibordered version is easier to work with since the adjacent triangles complete missing border cells, at least along horizontal or vertical edges. Diagonally it is sometimes necessary to overlap a diagonal cell with a top left corner cell. It is never allowed to abut two triangles prolonging a common top edge, which would imply one single large triangle on account of the run; otherwise the tiling is completely geometrical without any reference to the values of the cells they contain.
Superluminal shifts in the LCAU de Bruijn diagrams are a good source of regular lattices in the evolution of any automaton, especially because by taking the form of disjoint loops simple lattices result. However, other shifts and the cycle option can all give lattice examples. For rules such as 110 which have a textured background, which Cook calls the etherether, the likely candidates should be found anong the lattices of small period or cycle. From the point of view that Rule 110 depends on a triangular tiling of the plane, smaller triangles should be preferred, just on statistical grounds.