latticepattern

A lattice pattern refers to a periodic arrangement of points, lines, or motifs that fill space by repeating a basic unit at the lattice points of a geometric lattice. In mathematics, a lattice is the set of all integral linear combinations of a finite set of basis vectors in Euclidean space, describing how positions are repeated through translations. The pattern is defined by the choice of lattice (for example, square, rectangular, or hexagonal in 2D; cubic or hexagonal close-packed in 3D) and by the motif placed at each lattice point.

In two dimensions, lattice patterns produce tilings when motifs are connected or overlapped, resulting in designs such as grids, honeycomb-like networks, or diagonal grids. In crystallography, the lattice describes the periodic arrangement of atoms; in materials science, the lattice type influences properties such as symmetry, diffraction, and density.

Applications include textile design, architectural tiling, and computer graphics, where lattice-based repetition allows efficient pattern generation and scalable textures. Related concepts include Bravais lattices, which classify lattice types by symmetry, and tilings or wallpaper groups, which describe how motifs combine with lattice translations and rotations to create symmetric patterns.