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The seven lattice systems and their Bravais lattices in three dimensions. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (), [1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
The fourteen three-dimensional lattices, classified by lattice system, are shown above. The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices.
These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They [ clarification needed ] represent the maximum symmetry a structure with the given translational symmetry can have.
Bravais published a memoir about crystallography in 1847. A co-founder of the Société météorologique de France, he joined the French Academy of Sciences in 1854. Bravais also worked on the theory of observational errors, a field in which he is especially known for his 1846 paper "Mathematical analysis on the probability of errors of a point".
For the base-centered monoclinic lattice, the primitive cell has the shape of an oblique rhombic prism; [1] it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes.
In crystallography, a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices). [ 1 ]
Leave out the Bravais lattice type. Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.) Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.
The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).