Non-bijective translations on the triangular plane

In discrete and digital geometry, rotations with the composition of translations have been measured and examined carefully on the square and the hexagonal grids. The translation has never been considered individually because it obviously leads to the isometric translation on these grids. However, the triangular grid is not a point lattice, thus, it is worth to consider the translation itself. In this article, translations on the triangular grid are investigated and the vectors of bijective and non-bijective translations are specified.