Rhombille

Rhombille tiling is a plane tiling by congruent rhombi with angles of 60 and 120 degrees. Each rhombus has the same side length, and adjacent rhombi share full edges, producing a regular, repeating pattern.

One common way to construct the rhombille tiling is by overlaying two equilateral triangular lattices that are offset by half a unit along one direction. Equivalently, it can be viewed as subdividing a hexagonal tiling into rhombi, or as a 2D arrangement formed by three families of parallel lines at 0°, 60°, and 120°. Each cell is a single rhombus, and every vertex is the meeting point of three rhombi.

Key properties include its 3-regular vertex figure: three rhombi meet at every vertex, summing to 360 degrees. The tiling is periodic and possesses translational symmetry; it is also the dual tiling of the trihexagonal (3.6.3.6) tiling. In lattice terms, the rhombille net can be described as the rhombille lattice, a 2D structure formed from three directional families of lines.

Relation to other tilings: the rhombille tiling is closely connected to the hexagonal and triangular tilings as well as to the trihexagonal tiling through duality and shared geometric foundations. It is often used to illustrate tiling concepts because of its simple uniform cells and multiple symmetric viewpoints.

Applications and usage: the rhombille tiling appears in tiling theory, computational mesh generation, and geometric modeling. It also shows up in crystallography as a planar net and in design contexts where a regular, diamond-shaped tessellation is desired.

See also: trihexagonal tiling, hexagonal tiling, rhombille lattice, lattice.