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Upright square Simple |
diagonal square Centered |
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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 Z 2 {\displaystyle \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 , 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.
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:
p4, +, (442) | p4g, , (4*2) | p4m, , (*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). 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 |
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 | Wallpaper groups | ||||
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Schön. | Intl | Orb. | Cox. | ||
C4 | 4 | (44) | + | p4 (442) |
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D4 | 4mm | (*44) | p4m (*442) |
p4g (4*2) |
Crystal systems | |
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Seven 3D systems |
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Four 2D systems |