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Hexagonal tilings can be made with the identical {6,3} topology as the regular tiling (3 hexagons around every vertex). With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions. [2] Single-color (1-tile) lattices are parallelogon hexagons.
[1] [2] They're also called Shubnikov–Laves tilings after Aleksei Shubnikov. [3] John Conway called the uniform duals Catalan tilings , in parallel to the Catalan solid polyhedra. The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge.
A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90° rotated version.
“Tile companies have made great strides in suspending my disbelief that these aren’t actual stone, or other real materials,” says New York-based designer Daniel VanHall. The tech is also ...
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: square tiling , truncated square tiling , snub square tiling .
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