BravaisLattices

Bravaislattices are a theoretical generalization of Bravais lattices designed to describe periodic point sets with variable motifs attached to lattice points. Formally, they start from an underlying Bravais lattice L in Euclidean space, generated by a finite basis. A finite motif M is assigned to each lattice point by a motif function, yielding a full point set consisting of all translated motifs: for every lattice translation t in L and every motif point p in M, the point t + p is included. This construction preserves translational symmetry, so the overall pattern remains periodic. When M contains a single point and the motif is uniform, the Bravaislattice reduces to the ordinary Bravais lattice; larger motifs describe complex unit cells with multiple atoms or substructures.

Classification and study: Bravaislattices are classified by the type of underlying lattice (for example, cubic, tetragonal, or hexagonal in 3D) and by the symmetry and arrangement of the motif within the unit cell. Group-theoretic methods and lattice-symmetry analysis are used to determine possible equivalences and unfoldings of the structure.

Applications and status: The concept appears in discussions of generalized crystallography and mathematical tilings, where it helps model materials with internal substructure or composite unit cells. In mainstream crystallography, however, the standard Bravais lattice framework remains the primary tool.

See also Bravais lattice, crystal structure, unit cell, tiling, symmetry group.