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.
With respect to reflection axes there are three possibilities: 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.Ī 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. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.Īn 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. The square lattice's symmetry category is wallpaper group p4m. 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. They can conveniently be referred to as the upright square lattice and diagonal square lattice the latter is also called the centered square lattice. Two orientations of an image of the lattice are by far the most common. 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. It is the two-dimensional version of the integer lattice, denoted as Z 2. In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. 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. The vertices of all squares together with their centers form an upright square lattice.
2-dimensional integer lattice Square lattices
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