Abstract: Storing textures on orthogonal tensor product lattices is predominant in computer graphics, although it is known that their sampling efficiency is not optimal. In two dimensions, the hexagonal lattice provides the maximum sampling efficiency. However, handling these lattices is difficult, because they are not able to tile an arbitrary rectangular region and have an irrational basis. By storing textures on rank-1 lattices, we resolve both problems: Rank-1 lattices can closely approximate hexagonal lattices, while all coordinates of the lattice points remain integer. At identical memory footprint texture quality is improved as compared to traditional orthogonal tensor product lattices due to the higher sampling efficiency. We introduce the basic theory of rank-1 lattice textures and present an algorithmic framework which easily can be integrated into existing off-line and real-time rendering systems.