09-27-2021-1939 - Lattice and Bravais Lattice 1850 crystal lattice

 A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These lattice-likestructures all admit order-theoretic as well as algebraic descriptions.

https://en.wikipedia.org/wiki/Lattice_(order)

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850),^[1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by:

where the n_i are any integers and a_i are primitive translation vectors or primitive vectors which lie in different directions (not necessarily mutually perpendicular) and span the lattice. The choice of primitive vectors for a given Bravais lattice is not unique. A fundamental aspect of any Bravais lattice is that, for any choice of direction, the lattice will appear exactly the same from each of the discrete lattice points when looking in that chosen direction.

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) at each lattice point. The basis may consist of atoms, molecules, or polymer strings of solid matter.

Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 5 possible Bravais lattices in 2-dimensional space, and 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.^[2]

https://en.wikipedia.org/wiki/Bravais_lattice