Two-Dimensional Space Groups
The five basic lattice types
There are 17 space groups in the plane, but their unit cells fall into
one of five basic shapes as follows:
General Parallelogram lattice
+-----------+-----------+-----------+-----------+-----------+-----------+
/ / / / / /
/ / / / / /
/ / / / / /
--------+-----------+-----------+-----------+-----------+-----------+---
/ / / / / /
/ / / / / /
/ / / / / /
----+-----------+-----------+-----------+-----------+-----------+-------
/ / / / / /
/ / / / / /
/ / / / / /
+-----------+-----------+-----------+-----------+-----------+-----------+
General Rectangular Lattice
+-----------+-----------+-----------+-----------+-----------+-----------+
| | | | | | |
| | | | | | |
| | | | | | |
+-----------+-----------+-----------+-----------+-----------+-----------+
| | | | | | |
| | | | | | |
| | | | | | |
+-----------+-----------+-----------+-----------+-----------+-----------+
| | | | | | |
| | | | | | |
| | | | | | |
+-----------+-----------+-----------+-----------+-----------+-----------+
General Rhombic (Centered) Lattice
+ + + + +-------+-------+-------+-------+----
/ \ / \ / \ / \ /|\ /|\ /|\ /| |
/ \ / \ / \ / \ / | \ / | \ / | \ / | |
/ \ / \ / \ / \ / | \ / | \ / | \ / | |
+ + + + + | + | + | + | + | +
\ / \ / \ / \ / \ | / \ | / \ | / \ | |
\ / \ / \ / \ / \ | / \ | / \ | / \ | |
\ / \ / \ / \ / \|/ \|/ \|/ \| |
+ + + + +-------+-------+-------+-------+----
/ \ / \ / \ / \ /|\ /|\ /|\ /| |
/ \ / \ / \ / \ / | \ / | \ / | \ / | |
/ \ / \ / \ / \ / | \ / | \ / | \ / | |
+ + + + + | + | + | + | + | +
\ / \ / \ / \ / \ | / \ | / \ | / \ | |
\ / \ / \ / \ / \ | / \ | / \ | / \ | |
\ / \ / \ / \ / \|/ \|/ \|/ \| |
+ + + + +-------+-------+-------+-------+----
The simplest unit cell for this pattern is a rhombus, but the pattern also
has a rectangular structure. To bring out the rectangular pattern, this
lattice is often described as a rectangle with an extra point in the center.
Such a lattice is called centered.
Square Lattice
+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| | | | | | | | | |
| | | | | | | | | |
| | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+
Hexagonal Lattice
+ + + + + + + + +-----+-----+-----+---
/ / / /
/ / / /
+ + + + + + + + +-----+-----+-----+-----+
\ / \ / \ / \ /
\ / \ / \ / \ /
+ + + + + + + + + + + +
/ \ / \ / \ / \
/ \ / \ / \ / \
+ + + + + + + + + + + + +
\ / \ / \ / \ /
\ / \ / \ / \ /
+ + + + + + + + +-----+-----+-----+---
\ \ \ \
\ \ \ \
+ + + + + + + + + +-----+-----+-----+
Note that there are three equivalent ways to orient the unit cells.
The 17 Plane Space Groups
Parallelogram Lattices
1. p1
+ + + + + + + + + + + +
p p p p p p p p p p p p
+ + + + + + + + + + + +
p p p p p p p p p p p p
+ + + + + + + + + + +
p p p p p p p p p p p
+ + + + + + + + + + + +
2. p2
d d d d d d
+ + + + + + + + + + +
p p p p p p
+ + + + + + + + + + + +
d d d d d d
+ + + + + + + + + + +
p p p p p p
+ + + + + + + + + + + +
There are 2-fold axes at the + locations.
Rectangular Lattices
3. pm
p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b
---------------------------------------------------------------------
p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b
---------------------------------------------------------------------
p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b
---------------------------------------------------------------------
Parallel mirror planes
4. pg
p p p p p p p p p p p
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
p p p p p p p p p p p
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
p p p p p p p p p p p
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Parallel glide planes
5. pmm
+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|p | q|p | q|p | q|p | q|p | q|p | q|
+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|b | d|b | d|b | d|b | d|b | d|b | d|
+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|p | q|p | q|p | q|p | q|p | q|p | q|
+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|b | d|b | d|b | d|b | d|b | d|b | d|
+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
|p | q|p | q|p | q|p | q|p | q|p | q|
+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Perpendicular mirror planes
6. pmg
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
|p | q|p | q|p | q|p | q|p | q|p | q|
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
| d|b | d|b | d|b | d|b | d|b | d|b |
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
|p | q|p | q|p | q|p | q|p | q|p | q|
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
| d|b | d|b | d|b | d|b | d|b | d|b |
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
|p | q|p | q|p | q|p | q|p | q|p | q|
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
Perpendicular mirror and glide planes
>
7. pgg
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
|p d| |p d| |p d| |p d| |p d| |p d| |
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
| |b q| |b q| |b q| |b q| |b q| |b q|
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
|p d| |p d| |p d| |p d| |p d| |p d| |
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
| |b q| |b q| |b q| |b q| |b q| |b q|
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
|p d| |p d| |p d| |p d| |p d| |p d| |
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
| |b q| |b q| |b q| |b q| |b q| |b q|
+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+- - -+
Perpendicular glide planes. (p,b) and (d,q) are related by horizontal glides,
(p,q) and (b,d) by vertical glides. Note that the intersections of the
glide planes and the centers of the boxes outlined by the glide planes
are also two-fold symmetry axes.
Centered Lattices
8. cm
p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
p p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
p p p p p p p p p p p p
---------------------------------------------------------------------
b b b b b b b b b b b b
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Parallel mirror and glide planes. The pattern of stars on a 50-star flag
has this symmetry.
9. cmm
+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+
|p : | q|p : | : q|p : | : q|p : | : q|p : | : q|p : | : q|
+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+
|b : | : d|b : | : d|b : | : d|b : | : d|b : | : d|b : | : d|
+ -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- +
| : q|p : | : q|p : | : q|p : | : q|p : | : q|p : | : q|p : |
+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+
| : d|b : | : d|b : | : d|b : | : d|b : | : d|b : | : d|b : |
+ -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- +
|p : | : q|p : | : q|p : | : q|p : | : q|p : | : q|p : | : q|
+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+
|b : | : d|b : | : d|b : | : d|b : | : d|b : | : d|b : | : d|
+ -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- + -:- +
| : q|p : | : q|p : | : q|p : | : q|p : | : q|p : | : q|p : |
+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+--:--+
Alternating mirror and glide planes in both directions. All intersections
are also two-fold symmetry axes. Bricks in a wall have this symmetry.
Square Lattices
Trigonal Lattices
Hexagonal Lattices
Symmetry Elements of the Two-Dimensional Space Groups
