In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the square lattice is a type of
lattice in a two-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. It is the two-dimensional version of the
integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice (group), lattice in the Euclidean space whose lattice points are tuple, -tuples of integers. The two-dimensional integer lattice is also called the s ...
, denoted as . It is one of the five types of two-dimensional lattices as classified by their
symmetry groups; its symmetry group in
IUC notation as ,
Coxeter notation as , and
orbifold notation as .
Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice.
[.] They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a
checkerboard.
Symmetry
The square lattice's
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
category is
wallpaper group . A pattern with this lattice of
translational symmetry cannot have more, but may have less symmetry than the lattice itself.
An upright square lattice can be viewed as a diagonal square lattice with a mesh size that is √2 times as large, with the centers of the squares added. Correspondingly, after adding the centers of the squares of an upright square lattice one obtains a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice.
A pattern with 4-fold
rotational symmetry has a square lattice of 4-fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of
translational symmetry.
With respect to
reflection axes there are three possibilities:
*None. This is wallpaper group .
*In four directions. This is wallpaper group .
*In two
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
directions. This is wallpaper group . The points of intersection of the reflexion axes form a square grid which is as fine as, and oriented the same as, the square lattice of 4-fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection axes.
Crystal classes
The ''square lattice'' class names,
Schönflies notation,
Hermann-Mauguin notation,
orbifold notation,
Coxeter notation, and
wallpaper groups are listed in the table below.
See also
*
Centered square number
*
Euclid's orchard
*
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
*
Hexagonal lattice
The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an ...
*
Quincunx
*
Square tiling
References
{{DEFAULTSORT:Square Lattice
Euclidean geometry
Lattice points
Crystal systems