# Square lattice

Square lattices :
| Upright square Simple | diagonal square Centered |
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 ⁠$\mathbb {Z} ^{2}$⁠. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in IUC notation as p4m, Coxeter notation as [4,4], and orbifold notation as *442.
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 category is wallpaper group p4m. 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 p4.
* In four directions. This is wallpaper group p4m.
* In two perpendicular directions. This is wallpaper group p4g. 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.
| p4, [4,4]$^+$, (442) | p4g, [4,4$^+$], (4*2) | p4m, [4,4], (*442) |
| Wallpaper group p4, with the arrangement within a primitive cell of the 2- and 4-fold rotocenters (also applicable for p4g and p4m). Fundamental domain | Wallpaper group p4g. There are reflection axes in two directions, not through the 4-fold rotocenters. Fundamental domain | Wallpaper group p4m. There are reflection axes in four directions, through the 4-fold rotocenters. In two directions the reflection axes are oriented the same as, and as dense as, those for p4g, but shifted. In the other two directions they are linearly a factor √2 denser. Fundamental domain |

## 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.
| Geometric class, point group | Geometric class, point group | Geometric class, point group | Geometric class, point group | Wallpaper groups | Wallpaper groups |
| Schön. | Intl | Orb. | Cox. | Wallpaper groups | Wallpaper groups |
| C$_4$ | 4 | (44) | [4]$^+$ | p4 (442) | |
| D$_4$ | 4mm | (*44) | [4] | p4m (*442) | p4g (4*2) |