In geometry and crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation operations. A crystal is made up of one or more atoms (the basis) which is repeated at each lattice point. The crystal then looks the same when viewed from any of the lattice points. In all, there are 14 possible Bravais lattices that fill three-dimensional space. Table of Geometry, from the 1728 Cyclopaedia. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... Auguste Bravais (c. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

Development of the Bravais lattices

The 14 Bravais lattices are arrived at by combining one of the seven crystal systems (or axial systems) with one of the lattice centerings. In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

The lattice centerings are:

• Primitive centering (P): lattice points on the cell corners only
• Body centered (I): one additional lattice point at the center of the cell
• Face centered (F): one additional lattice point at center of each of the faces of the cell
• Centered on a single face (A, B or C centering): one additional lattice point at the center of one of the cell faces.

Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centered lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.

Crystal system	Bravais lattices
triclinic	P
monoclinic	P	C
orthorhombic	P	C	I	F
tetragonal	P	I
rhombohedral (trigonal)	P
hexagonal	P
cubic	P	I	F

In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... Triclinic crystal structure File links The following pages link to this file: User:DrBob/Figures Crystal structure Triclinic Categories: GFDL images ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... Monoclinic crystal structure. ... Monoclinic base-centred crystal structure File links The following pages link to this file: User:DrBob/Figures Crystal structure Monoclinic Categories: GFDL images ... In crystallography, the orthorhombic crystal system is one of the 7 lattice point groups. ... Orthorhombic crystal structure. ... Orthorhombic base-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Orthorhombic Categories: GFDL images ... Orthorhombic body-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Orthorhombic Categories: GFDL images ... Orthorhombic face-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Orthorhombic Categories: GFDL images ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... Tetragonal crystal structure. ... Tetragonal body-centred crystal File links The following pages link to this file: User:DrBob/Figures Crystal structure Tetragonal Categories: GFDL images ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the 7 lattice point groups. ... Rhombohedral crystal structure. ... In crystallography, the hexagonal crystal system is one of the 7 lattice point groups. ... Image File history File links Hexagonal. ... In crystallography, the cubic crystal system (or isometric crystal system) is the most symmetric of the 7 crystal systems. ... Cubic crystal structure. ... Image File history File links Cubic-body-centered. ... Image File history File links Cubic-face-centered. ...

The volume of the unit cell can be calculated by evaluating where , and are the lattice vectors, the volumes of the Bravais lattices are given below:

In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ... In crystallography, the orthorhombic crystal system is one of the 7 lattice point groups. ... In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. ... In crystallography, the rhombohedral (or trigonal) crystal system is one of the 7 lattice point groups. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... The cubic crystal system is a crystal system where the unit cell is in the shape of a cube. ...

See also

• translational symmetry
• lattice (group)
• classification of lattices