Square lattice

{{Short description|2-dimensional integer lattice}}

Image:Square Lattice Tiling.svg. The vertices of all squares together with their centers form an upright square lattice. 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.]]

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 {{tmath|\mathbb{Z}^2}}.{{citation|title=Sphere Packings, Lattices and Groups|first1=John|last1=Conway|author1-link=John Horton Conway|first2=Neil J. A.|last2=Sloane|author2-link=Neil Sloane|publisher=Springer|year=1999|isbn=9780387985855|page=106|url=https://books.google.com/books?id=upYwZ6cQumoC&pg=PA106}}. It is one of the five types of two-dimensional lattices as classified by their symmetry groups;{{citation|title=The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space|volume=200|series=Progress in Mathematics|first1=Martin|last1=Golubitsky|author1-link=Marty Golubitsky|first2=Ian|last2=Stewart|author2-link=Ian Stewart (mathematician)|publisher=Springer|year=2003|isbn=9783764321710|page=129|url=https://books.google.com/books?id=0HpyrroR9REC&pg=PA129}}. its symmetry group in IUC notation as {{math|p4m}},{{citation|title=Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature|edition=2nd|first1=Michael|last1=Field|first2=Martin|last2=Golubitsky|author2-link=Marty Golubitsky|publisher=SIAM|year=2009|isbn=9780898717709|page=47|url=https://books.google.com/books?id=tu2Hnnc-b3YC&pg=PA47}}. Coxeter notation as {{math|[4,4]}},{{citation|title=Quadratic integers and Coxeter groups|doi=10.4153/CJM-1999-060-6|journal=Canadian Journal of Mathematics|volume=51|issue=6|year=1999|pages=1307–1336|first1=Norman W.|last1= Johnson|author1-link=Norman Johnson (mathematician)|first2=Asia Ivić|last2=Weiss|doi-access=free}}. See in particular the top of p. 1320. and orbifold notation as {{math|*442}}.{{citation |title=Handbook of Discrete and Computational Geometry |edition=2nd |series=Discrete Mathematics and Its Applications |editor1-first=Jacob E. |editor1-last=Goodman |editor1-link=Jacob E. Goodman |editor2-first=Joseph |editor2-last=O'Rourke |editor2-link=Joseph O'Rourke (professor) |publisher=CRC Press |year=2004 |isbn=9781420035315 |pages=53–72 |chapter=Tilings |first1=Doris |last1=Schattschneider |author1-link=Doris Schattschneider |first2=Marjorie |last2=Senechal |author2-link=Marjorie Senechal}}. See in particular the table on [https://books.google.com/books?id=QS6vnl8WlnQC&pg=PA62 p. 62] relating IUC notation to orbifold notation.

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.{{citation|title=Numbers and Symmetry: An Introduction to Algebra|first1=Bernard L.|last1=Johnston|first2=Fred|last2=Richman|publisher=CRC Press|year=1997|isbn=9780849303012|page=159|url=https://books.google.com/books?id=koUfrlgsmUcC&pg=PA159}}. 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.