Editing Lattice (group) (section)

==Lattices in two dimensions: detailed discussion==
[[File:2d-bravais.svg|thumb|Five lattices in the Euclidean plane]]
There are five 2D lattice types as given by the [[crystallographic restriction theorem]]. Below, the [[wallpaper group]] of the lattice is given in [[IUC notation|IUCr notation]], [[Orbifold notation]], and [[Coxeter notation]], along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. A [[List of planar symmetry groups#Wallpaper groups|full list of subgroups]] is available. For example, below the hexagonal/triangular lattice is given twice, with full 6-fold and a half 3-fold reflectional symmetry. If the symmetry group of a pattern contains an ''n''-fold rotation then the lattice has ''n''-fold symmetry for even ''n'' and 2''n''-fold for odd ''n''.
{| class=wikitable width=780
|-
!cmm, (2*22), [&infin;,2<sup>+</sup>,&infin;]
!p4m, (*442), [4,4]
!p6m, (*632), [6,3]
|- valign=top align=center
|[[File:Rhombic Lattice.svg|150px]][[File:Wallpaper group diagram cmm.svg|100px]]<BR>'''[[rhombus|rhombic]] lattice'''<BR>also '''centered [[rectangle|rectangular]] lattice'''<BR>''isosceles triangular''
|[[File:SquareLattice.svg|150px]][[File:Wallpaper group diagram p4m square.svg|75px]]<BR>'''[[square lattice]]'''<BR>''right isosceles triangular''
|[[File:Equilateral Triangle Lattice.svg|150px]][[File:Wallpaper group diagram p6m.svg|100px]]<BR>'''[[hexagonal lattice]]'''<BR>(equilateral triangular lattice)
|-
!pmm, *2222, [&infin;,2,&infin;]
!p2, 2222, [&infin;,2,&infin;]<sup>+</sup>
!p3m1, (*333), [3<sup>[3]</sup>]
|- valign=top align=center
|[[File:Rectangular Lattice.svg|150px]][[File:Wallpaper group diagram pmm.svg|100px]]<BR>'''[[rectangular lattice]]'''<BR>also '''centered rhombic lattice'''<BR>''right triangular''
|[[File:Oblique Lattice.svg|150px]][[File:Wallpaper group diagram p2.svg|100px]]<BR>'''[[oblique lattice]]'''<BR>''scalene triangular''
|[[File:Equilateral Triangle Lattice.svg|150px]][[File:Wallpaper group diagram p3m1.svg|100px]]<BR>'''equilateral [[triangle|triangular]] lattice'''<BR>(hexagonal lattice)
|}

For the classification of a given lattice, start with one point and take a nearest second point. For the third point, not on the same line, consider its distances to both points. Among the points for which the smaller of these two distances is least, choose a point for which the larger of the two is least. (Not [[Logical equivalence|logically equivalent]] but in the case of lattices giving the same result is just "Choose a point for which the larger of the two is least".)

The five cases correspond to the [[triangle]] being equilateral, right isosceles, right, isosceles, and scalene. In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle. This depends on the smaller angle of the rhombus being less than 60° or between 60° and 90°.

The general case is known as a [[period lattice]]. If the vectors '''p''' and '''q''' generate the lattice, instead of '''p''' and '''q''' we can also take '''p''' and '''p'''-'''q''', etc. In general in 2D, we can take ''a'' '''p''' + ''b'' '''q''' and ''c'' '''p''' + ''d'' '''q''' for integers ''a'',''b'', ''c'' and ''d'' such that ''ad-bc'' is 1 or -1. This ensures that '''p''' and '''q''' themselves are integer linear combinations of the other two vectors.  Each pair '''p''', '''q''' defines a parallelogram, all with the same area, the magnitude of the [[cross product]]. One parallelogram fully defines the whole  object. Without further symmetry, this parallelogram is a [[fundamental pair of periods|fundamental parallelogram]].

[[File:ModularGroup-FundamentalDomain.svg|thumb|right|The [[fundamental domain]] of the [[period lattice]].]]
The vectors '''p''' and '''q''' can be represented by [[complex number]]s. Up to size and orientation, a pair can be represented by their quotient. Expressed geometrically: if two lattice points are 0 and 1, we consider the position of a third lattice point. Equivalence in the sense of generating the same lattice is represented by the [[modular group]]: <math>T: z\mapsto z+1</math> represents choosing a different third point in the same grid, <math>S: z\mapsto -1/z</math> represents choosing a different side of the triangle as reference side 0–1, which in general implies changing the scaling of the lattice, and rotating it. Each "curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonical representation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to each other; duplication is avoided by including only half of the boundary. The rhombic lattices are represented by the points on its boundary, with the hexagonal lattice as vertex, and ''i'' for the square lattice. The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammatic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.