in geometry and crystallography, a bravais lattice, named after auguste bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by:
where ni are any integers and ai are primitive vectors which lie in different directions (not necessarily mutually perpendicular) and span the lattice. this discrete set of vectors must be closed under vector addition and subtraction. for any choice of position vector r, the lattice looks exactly the same.
when the discrete points are atoms, ions, or polymer strings of solid matter, the bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers. a crystal is made up of a periodic arrangement of one or more atoms (the basis, or motif) repeated at each lattice point. consequently, the crystal looks the same when viewed from any equivalent lattice point, namely those separated by the translation of one unit cell.
two bravais lattices are often considered equivalent if they have isomorphic symmetry groups. in this sense, there are 14 possible bravais lattices in three-dimensional space. the 14 possible symmetry groups of bravais lattices are 14 of the 230 space groups. in the context of the space group classification, the bravais lattices are also called bravais classes, bravais arithmetic classes, or bravais flocks.