Square lattice
| File:Square Lattice.svg | |
| Upright square Simple |
diagonal square Centered |
|---|---|
File:Square Lattice Tiling.svg
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 Z2.[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 p4m,[3] Coxeter notation as [4,4],[4] and orbifold notation as *442.[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.
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, [4,4]+, (442) | p4g, [4,4+], (4*2) | p4m, [4,4], (*442) |
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| 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
References
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