Rhombille tiling
| Rhombille tiling | |
|---|---|
 |
|
| Type | Laves tiling |
| Faces | 60°–120° rhombus |
| Coxeter diagram |
    
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| Symmetry group | p6m, [6,3], *632 p3m1, [3[3]], *333 |
| Rotation group | p6, [6,3]+, (632) p3, [3[3]]+, (333) |
| Dual polyhedron | Trihexagonal tiling |
| Face configuration | V3.6.3.6
 |
| Properties | edge-transitive face-transitive |
In geometry, the rhombille tiling, also known as tumbling blocks,reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles.
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.
The diagonals of each rhomb are in the ratio 1:√3. This is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the face configuration for monohedral tilings it is denoted [3.6.3.6].
It is also one of 56 possible isohedral tilings by quadrilaterals, and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points (x,y,z) with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, and more strongly such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube.
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