Triangular tiling

The triangular tiling, triangular grid, triangular tessellation or triangular lattice (Schläfli symbol {3,6}) is one of the three possible regular tilings of the plane, alongside the square tiling and hexagonal tiling. It is constructed with six triangles being placed at each vertex.

Of the three regular 2D tilings, it is by far the least investigated, with very little native software support. Native support for triangular rules is offered by some general-purpose cellular automaton simulation programs, although said support is seldom without flaw.

Coordinates, directions and displacements

While Cartesian coordinates can be used to describe the positions of triangles, this does not preserve symmetry well, as it effectively treats the triangular tiling as a square tiling and fails to consider its unique symmetries. While there exist many coordinate systems which can be used to describe the coordinates of triangles in a symmetrical way, such as cube coordinates, none seem to be in use in cellular automaton simulation programs.

Orthogonal and diagonal directions

While the existence of distinct orthogonal and diagonal directions on a triangular grid is not immediately as obvious as on a square grid, they are distinct and have been defined.^[1] By analogy to the hexagonal grid, if we consider a vertex of the triangular grid, the six edges radiating out from the vertex are considered as pointing in the six orthogonal directions, whereas the six rays directly between these (passing through cell centers) are considered to be the six diagonal directions.

Neighbourhoods on the triangular tiling

For what is considered to be a range of 1, the triangular tiling has the greatest diversity of neighbourhoods: there are effectively three types of cells adjacent to the central cell, giving a total of seven unique neighbourhoods (excluding the null case). These are listed on the Triangular neighbourhood page. However, the six-cell "outer" neighbourhood is effectively trivial, as it emulates two non-interacting hexagonal grids.^[2] The "radiation" neighbourhood similarly splits the grid into four non-interacting grids which can be described with the simpler "edges" neighbourhood.