An upright square lattice (left) and a diagonal square lattice (right).

Upright square tiling. For each color the centers of the squares of that color form a diagonal square lattice which is in linear scale √2 times as large as the upright square lattice of vertices. The vertices of all squares together with their centers form a diagonal square lattice which is in linear scale √2 times as small as the upright square lattice of vertices.

In mathematics, the square lattice is one of the five two-dimensional lattice types. It is the two-dimensional version of the integer lattice.

Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as "upright square lattice" and "diagonal square lattice". They differ by an angle of 45°.

Symmetry

The square latice'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 we have 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.

Wallpaper group p4, with the arrangement within a primitive cell of the 2- and 4-fold rotocenters (also applicable for p4g and p4m). A fundamental domain is indicated in yellow.	Wallpaper group p4g. There are reflection axes in two directions, not through the 4-fold rotocenters.	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.

See also

• square tiling
• hexagonal lattice
• symmetry combinations
• centered square number
• Gaussian integer