Square lattice

Square lattice

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°.


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.

ee also

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

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