Square lattice

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Square lattices
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)
Wallpaper group diagram p4 square.svg Wallpaper group diagram p4g square.svg Wallpaper group diagram p4m square.svg
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

  1. Conway, John; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Springer, p. 106, ISBN 9780387985855<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
  2. Golubitsky, Martin; Stewart, Ian (2003), The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, 200, Springer, p. 129, ISBN 9783764321710<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
  3. Field, Michael; Golubitsky, Martin (2009), Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature (2nd ed.), SIAM, p. 47, ISBN 9780898717709<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.
  4. Johnson, Norman W.; Weiss, Asia Ivić (1999), "Quadratic integers and Coxeter groups", Canadian Journal of Mathematics, 51: 1307–1336, doi:10.4153/CJM-1999-060-6<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. See in particular the top of p. 1320.
  5. 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. See in particular the table on p. 62 relating IUC notation to orbifold notation.
  6. Johnston, Bernard L.; Richman, Fred (1997), Numbers and Symmetry: An Introduction to Algebra, CRC Press, p. 159, ISBN 9780849303012<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>.