Symmetry Considerations and Examples
A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. As a group (dropping its geometric structure) a lattice is a finitely-generated free abelian group.
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translate of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense.
A simple example of a lattice in is the subgroup . A more complicated example is the Leech lattice, which is a lattice in . The period lattice in is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions.
Read more about this topic: Lattice (group)
Famous quotes containing the words symmetry and/or examples:
“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)