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The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones. [4] Each vertex has edges evenly spaced around it.
One approach is to color the vertices (with two colors, e.g., black and white) and require that adjacent tiles have matching vertices. [32] Another is to use a pattern of circular arcs (as shown above left in green and red) to constrain the placement of tiles: when two tiles share an edge in a tiling, the patterns must match at these edges. [21]
Following Grünbaum and Shephard (section 1.3), a tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first ...
The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks. [ 18 ] A normal tiling is a tessellation for which every tile is topologically equivalent to a disk , the intersection of any two tiles is a connected set or the empty set , and all tiles are uniformly bounded .
Robinson proves these tiles must form this structure inductively; in effect, the tiles must form blocks which themselves fit together as larger versions of the original tiles, and so on. This idea – of finding sets of tiles that can only admit hierarchical structures – has been used in the construction of most known aperiodic sets of tiles ...
It is a square tiling with each edge chamfered into new hexagonal faces. It can also be seen as the intersection of two truncated square tilings with offset positions. And its appearance is similar to a truncated square tiling, except only half of the vertices have been truncated, leading to its descriptive name semitruncated square tiling.
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