Derivation of Lattice Systems in 2D
Lattice systems in 2D are identified based on their rotational symmetry elements. Watch the lattice naturally emerge through rotation and translation.
The 2D Lattices
Select a lattice type to animate its geometric derivation.
Step 1 of 4
Start with a single lattice point O.
The 17 Plane Groups Builder
Plane groups are formed by adding symmetry-compatible motifs to the lattice arrays. Watch the point group symmetry methodically build the fundamental basis step-by-step around the origin before replicating across space.
Sequential Basis Construction
Methodically construct the basis and map all structural symmetry.
Step 1 of X
Construct the lattice array.
Hermann-Mauguin Conventions
The primary, secondary, and tertiary positions of the three-component H-M notations depend on the lattice under consideration. As per the International Tables for Crystallography conventions for 2D lattices, the directions are assigned as follows:
| Symmetry direction (position in Hermann–Mauguin symbol) | |||
|---|---|---|---|
| Lattice | Primary | Secondary | Tertiary |
| Two dimensions | |||
| Oblique | Rotation point in plane |
||
| Rectangular | $[10]$ | $[01]$ | |
| Square | $\left\{ \begin{matrix} [10] \\ [01] \end{matrix} \right\}$ | $\left\{ \begin{matrix} [1\bar{1}] \\ [11] \end{matrix} \right\}$ | |
| Hexagonal | $\left\{ \begin{matrix} [10] \\ [01] \\ [\bar{1}\bar{1}] \end{matrix} \right\}$ | $\left\{ \begin{matrix} [1\bar{1}] \\ [12] \\ [\bar{2}\bar{1}] \end{matrix} \right\}$ | |
In a plane (2D) or 3D space group stereographic projection diagram, the origin, by convention, is assumed to be at the top left of the unit cell, and the basis vectors used for Miller Indices originate from here.
Visualising $p31m$ vs $p3m1$
p31m
Primary: 3-fold rotation axis.
Secondary Direction $[10]$: Contains NO mirror plane perpendicular to it. Denoted by 1.
Tertiary Direction $[1\bar{1}]$: Contains a mirror plane perpendicular to it. Denoted by m.
p3m1
Primary: 3-fold rotation axis.
Secondary Direction $[10]$: Contains a mirror plane perpendicular to it. Denoted by m.
Tertiary Direction $[1\bar{1}]$: Contains NO mirror plane perpendicular to it. Denoted by 1.
Space Group Flowchart
Flowchart to determine 2D space (plane) group
-
Structure
-
6-fold axis
-
Has
mirrors?- Yes
- p6m
- No
- p6
-
-
4-fold axis
-
Has
mirrors?-
Yes
-
Axes on
mirrors?- Yes
- p4m
- No
- p4g
-
- No
- p4
-
-
-
3-fold axis
-
Has
mirrors?-
Yes
-
All axes on
mirrors?- Yes
- p3m1
- No
- p31m
-
- No
- p3
-
-
-
2-fold axis
-
Has
mirrors?-
Yes
-
Has
orthogonal
mirrors?-
Yes
-
Has
rotations
off mirrors?- Yes
- cmm
- No
- pmm
-
- No
- pmg
-
-
-
No
-
Has glides?
- Yes
- pgg
- No
- p2
-
-
-
-
1-fold axis
-
Has
mirrors?-
Yes
-
Has glides?
- Yes
- cm
- No
- pm
-
-
No
-
Has glides?
- Yes
- pg
- No
- p1
-
-
-
-
Orthorhombic Notations
For crystallographic point groups compatible with the orthorhombic system, the rule slightly changes. For example, $2mm$ is often written as mm2 because the 2-fold rotation is taken to be in the tertiary direction (along the $c$-axis) when drawing stereographic projections.
Key Pointers & Reminders
-
Order of a point group = Number of symmetry elements present.
-
While solving plane group identification:
- Identify the order of rotation correctly.
- Ensure that the unit cell is identified correctly.
- Ensure that ALL symmetry elements for a particular plane group are present.
-
Subgroups: If a point group can be added to a compatible lattice, naturally, its subgroups will be present in that lattice too.
Group-Subgroup Relationships (32 Point Groups)
Did It Sink In? - 8-9
Test your mastery of 2D lattice derivation, implicit symmetry generation, and the mathematical rules governing the 17 plane groups.
2D Symmetry Solver
Put your knowledge to the test! Use the flowchart logic to deduce the plane groups of 17 randomized 2D materials in our interactive WebGL sandbox.