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Rectangular lattices
Primitive Centered
pmm cmm

The rectangular lattice and centered rectangular lattice (or rhombic lattice) constitute two of the five two-dimensional Bravais lattice types.[1] The symmetry categories of these lattices are wallpaper groups pmm and cmm respectively. The conventional translation vectors of the rectangular lattices form an angle of 90° and are of unequal lengths.

Bravais lattices

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There are two rectangular Bravais lattices: primitive rectangular and centered rectangular (or rhombic).

Rectangular vs rhombic unit cells for the 2D rectangular lattices.
Bravais lattice Rectangular Centered rectangular
Pearson symbol op oc
Standard unit cell
Rhombic unit cell

The primitive rectangular lattice can also be described by a centered rhombic unit cell, while the centered rectangular lattice can also be described by a primitive rhombic unit cell. Note that the length in the lower row is not the same as in the upper row. For the first column above, of the second row equals of the first row, and for the second column it equals .

Crystal classes

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The rectangular 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 Arithmetic
class 		Wallpaper groups
Schön. Intl Orb. Cox.
D1 m (*) [ ] Along pm
(**) 		pg
(××)
Between cm
(*×) 		 
D2 2mm (*22) [2] Along pmm
(*2222) 		pmg
(22*)
Between cmm
(2*22) 		pgg
(22×)

References

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  1. ^ Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18.


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Lattice (group)

Coordinate-wise addition or subtraction of two points in the lattice produces another lattice point. The lattice points are all separated by some minimum distance

Wallpaper group

square (square lattice, itself p4m). In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm)

Orthorhombic crystal system

Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with

Bravais lattice

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of

Rhombus

with exponent 1. One of the five 2D lattice types is the rhombic lattice, also called centered rectangular lattice. Rhombi can tile the 2D plane edge-to-edge

Tetragonal crystal system

Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square

Unit cell

two-dimensional Bravais lattices are represented using conventional primitive cells, as shown below. The centered rectangular lattice also has a primitive

Crystal system

(oblique, rectangular, square, hexagonal), four crystal families (oblique, rectanguar, square, hexagonal), and four lattice systems (oblique, rectangular, square