Square lattice: Difference between revisions
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[[Image:Square Lattice Tiling.svg|thumb|Upright [[square tiling]]. 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.]] | [[Image:Square Lattice Tiling.svg|thumb|Upright [[square tiling]]. 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 (group)|lattice]] in a two-dimensional [[Euclidean space]]. It is the two-dimensional version of the [[integer lattice]], denoted as {{tmath|\mathbb{Z}^2}}.<ref>{{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= | In [[mathematics]], the '''square lattice''' is a type of [[Lattice (group)|lattice]] in a two-dimensional [[Euclidean space]]. It is the two-dimensional version of the [[integer lattice]], denoted as {{tmath|\mathbb{Z}^2}}.<ref>{{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=[https://books.google.com/books?id=upYwZ6cQumoC&pg=PA106 106]|title-link=Sphere Packings, Lattices and Groups}}.</ref> It is one of the five types of two-dimensional lattices as classified by their [[symmetry group]]s;<ref>{{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}}.</ref> its symmetry group in [[IUC notation#Symmetry notation|IUC notation]] as {{math|[[Wallpaper group#Group p4m|p4m]]}},<ref>{{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}}.</ref> [[Coxeter notation]] as {{math|[4,4]}},<ref>{{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.</ref> and [[orbifold notation]] as {{math|*442}}.<ref>{{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.</ref> | ||
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'''.<ref>{{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}}.</ref> 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]]. | 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'''.<ref>{{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}}.</ref> 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]]. | ||
Latest revision as of 08:29, 26 June 2025
| File:Square Lattice.svg | |
| 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 Template:Tmath.[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 Template:Math,[3] Coxeter notation as Template:Math,[4] and orbifold notation as Template:Math.[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 Template:Math. 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 Template:Math.
- In four directions. This is wallpaper group Template:Math.
- In two perpendicular directions. This is wallpaper group Template:Math. 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.
| Template:Math | Template:Math | Template:Math |
|---|---|---|
| File:Wallpaper group diagram p4 square.svg | File:Wallpaper group diagram p4g square.svg | File:Wallpaper group diagram p4m square.svg |
| Wallpaper group Template:Math, with the arrangement within a primitive cell of the 2- and 4-fold rotocenters (also applicable for Template:Math and Template:Math). <templatestyles src="Legend/styles.css" /> Fundamental domain
|
Wallpaper group Template:Math. There are reflection axes in two directions, not through the 4-fold rotocenters. <templatestyles src="Legend/styles.css" /> Fundamental domain
|
Wallpaper group Template:Math. 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 Template:Math, but shifted. In the other two directions they are linearly a factor √2 denser. <templatestyles src="Legend/styles.css" /> Fundamental domain
|
Crystal classes
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
References
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- ↑ Script error: No such module "citation/CS1".. See in particular the top of p. 1320.
- ↑ Script error: No such module "citation/CS1".. See in particular the table on p. 62 relating IUC notation to orbifold notation.
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