Square Lattice - Symmetry

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 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.
p4, +, (442) p4g, (4*2) p4m, (*442)
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.

Read more about this topic:  Square Lattice

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