Rectangular lattice

Rectangular lattices

File:Rectangular Lattice.svg	File:Rhombic Lattice.svg
Primitive	Centered
File:Wallpaper group diagram pmm.svg	File:Wallpaper group diagram cmm.svg
pmm	cmm

The rectangular lattice and centered rectangular lattice (or rhombic lattice) constitute two of the five two-dimensional Bravais lattice types. 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

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

File:Rectangular unit cells.svg

Rectangular vs rhombic unit cells for the 2D rectangular lattices.

Bravais lattice	Rectangular	Centered rectangular
Pearson symbol	op	oc
Standard unit cell	File:2d op rectangular.svg	File:2d oc rectangular.svg
Rhombic unit cell	File:2d op rhombic.svg	File:2d oc rhombic.svg

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 $a$ in the lower row is not the same as in the upper row. For the first column above, $a$ of the second row equals $\sqrt{a^2+b^2}$ of the first row, and for the second column it equals $\frac{1}{2}\sqrt{a^2+b^2}$.

Crystal classes

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×)