In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as .[1] It is one of the five types of two-dimensional lattices as classified by their symmetry groups;[2] its symmetry group in IUC notation as,[3] Coxeter notation as,[4] and orbifold notation as .[5]
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.[6] 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.
The square lattice's symmetry 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:
The square lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
| Geometric class, point group | Wallpaper groups | |||||
|---|---|---|---|---|---|---|
| Cox. | ||||||
| C4 | 4 | (44) | [4]+ | p4 (442) | ||
| D4 | 4mm | (*44) | [4] | p4m (*442) | p4g (4*2) | |