Square lattice
| Upright square Simple |
Diagonal square Centered |
|---|---|
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 {\displaystyle \mathbb {Z} ^{2}}.[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
[edit ]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:
- 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) |
|---|---|---|
| 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
|
Crystal classes
[edit ]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 | ||||
|---|---|---|---|---|---|
| Schön. | Intl | Orb. | Cox. | ||
| C4 | 4 | (44) | [4]+ | p4 (442) |
|
| D4 | 4mm | (*44) | [4] | p4m (*442) |
p4g (4*2) |
See also
[edit ]References
[edit ]- ^ Conway, John; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Springer, p. 106, ISBN 9780387985855 .
- ^ Golubitsky, Martin; Stewart, Ian (2003), The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, vol. 200, Springer, p. 129, ISBN 9783764321710 .
- ^ Field, Michael; Golubitsky, Martin (2009), Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature (2nd ed.), SIAM, p. 47, ISBN 9780898717709 .
- ^ Johnson, Norman W.; Weiss, Asia Ivić (1999), "Quadratic integers and Coxeter groups", Canadian Journal of Mathematics, 51 (6): 1307–1336, doi:10.4153/CJM-1999-060-6 . See in particular the top of p. 1320.
- ^ Schattschneider, Doris; Senechal, Marjorie (2004), "Tilings", in Goodman, Jacob E.; O'Rourke, Joseph (eds.), Handbook of Discrete and Computational Geometry, Discrete Mathematics and Its Applications (2nd ed.), CRC Press, pp. 53–72, ISBN 9781420035315 . See in particular the table on p. 62 relating IUC notation to orbifold notation.
- ^ Johnston, Bernard L.; Richman, Fred (1997), Numbers and Symmetry: An Introduction to Algebra, CRC Press, p. 159, ISBN 9780849303012 .
