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Rectangular lattice

2-dimensional lattice

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Source updated September 10, 2025

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. 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.

01Bravais lattices

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

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 $a$ 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 ${\frac {1}{2}}{\sqrt {a^{2}+b^{2}}}$ .

02Crystal 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.	Arithmetic class	Wallpaper groups
D₁	m	(*)	[ ]	Along	pm (**)	pg (××)
D₁	m	(*)	[ ]	Between	cm (*×)
D₂	2mm	(*22)	[2]	Along	pmm (*2222)	pmg (22*)
D₂	2mm	(*22)	[2]	Between	cmm (2*22)	pgg (22×)

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Sources and credits

This article is adapted from the Wikipedia article “Rectangular lattice”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Rectangular Lattice.svg by Jim.belk, Public domain
Rectangular unit cells.svg by User:Prolineserver, CC BY-SA 3.0

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