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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.
This symmetry will generally preserve only a patch of tiles around the center point, but the patch can be very large: Conway and Penrose proved that whenever the colored curves on the P2 or P3 tilings close in a loop, the region within the loop has pentagonal symmetry, and furthermore, in any tiling, there are at most two such curves of each ...
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.
Each hexagon of one tiling surrounds two vertices of the other tiling, and is divided by the hexagons of the other tiling into four of the pentagons in the Cairo tiling. [4] Infinitely many different pentagons can form Cairo tilings, all with the same pattern of adjacencies between tiles and with the same decomposition into hexagons, but with ...
With a final vertex 3 4.6, 4 more contiguous equilateral triangles and a single regular hexagon. However, this notation has two main problems related to ambiguous conformation and uniqueness [2] First, when it comes to k-uniform tilings, the notation does not explain the relationships between the vertices. This makes it impossible to generate a ...
The rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling. It can also be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi.