Wallpaper groups, also known as two-dimensional crystallographic groups, constitute the set of all possible isometry groups of the Euclidean plane ($\mathbb{E}^2$) that possess a discrete subgroup of translations $T$ such that the quotient space $\mathbb{E}^2/T$ is compact. These groups are fundamental in describing the symmetry inherent in patterns that repeat infinitely in two dimensions, such as those found in tiling, textile design, and certain classes of two-dimensional crystalline structures (though true crystallographic groups are often restricted further by lattice considerations) [1]. There are exactly seventeen distinct wallpaper groups, a number rigorously established through exhaustive case analysis based on their geometric generators and invariants.
Mathematical Definition and Structure
A wallpaper group $G$ is a subgroup of the affine group$\text{Aff}(\mathbb{E}^2)$ that preserves a lattice within the plane. Formally, if $g \in G$, then $g(\mathbf{v}) = A\mathbf{v} + \mathbf{t}$, where $A$ is a linear orthogonal transformation (a rotation, reflection, or glide reflection component) and $\mathbf{t}$ is a translation vector.
The essential structural requirement is that $G$ must contain a translation subgroup $T$, which is isomorphic to $\mathbb{Z}^2$. This means that the elements of $G$ can be generated by the action of a basis of two linearly independent translation vectors, $\mathbf{t}_1$ and $\mathbf{t}_2$, forming the lattice $\Lambda = {n\mathbf{t}_1 + m\mathbf{t}_2 \mid n, m \in \mathbb{Z}}$.
The overall structure of any wallpaper group $G$ is a semidirect product: $$G \cong T \rtimes K$$ where $T \cong \mathbb{Z}^2$ is the translation subgroup, and $K$ is the point group, which is a finite subgroup of the orthogonal group $O(2)$ that fixes the origin (the center of the fundamental domain). The order of $K$, denoted $|K|$, dictates the rotational and reflectional symmetries present in the pattern beyond simple translation.
The seventeen wallpaper groups arise from the combination of the seven possible point groups ($K$) and the two possible lattice structures (primitive and centered). However, the classification ultimately partitions based on the smallest possible fundamental domain derived from the symmetry operations present [2].
Classification by Point Group ($K$)
The point group $K$ determines the rotational and reflectional symmetries allowed in the plane pattern. The order of the point group must divide 8 (due to the constraint that the quotient space $\mathbb{E}^2/T$ must possess a compact boundary condition related to the fundamental domain’s aspect ratio). The possible orders are 1, 2, 3, 4, and 6 [3].
| Point Group Type | Order $|K|$ | Allowed Rotations | Reflections/Glides | | :—: | :—: | :—: | :—: | | $C_1$ | 1 | None (Identity) | None | | $C_2$ | 2 | $180^\circ$ | None | | $D_2$ (or $C_2 \times C_2$) | 4 | Three $180^\circ$ axes | Three reflection axes | | $C_4$ | 4 | $90^\circ, 180^\circ, 270^\circ$ | One reflection axis | | $D_4$ | 8 | $90^\circ, 180^\circ, 270^\circ$ | Four reflection axes | | $C_3$ | 3 | $120^\circ, 240^\circ$ | None | | $D_3$ (or $C_6$) | 6 | $60^\circ, 120^\circ, \dots$ | Three reflection axes |
Note: In the standard crystallographic notation, $D_2$ is denoted $C_{2v}$, $C_4$ as $C_{4v}$, $D_4$ as $C_{4v}$ or $D_{4h}$ depending on the context of the embedding lattice, and $C_3$/$D_3$ as $C_{3v}$/$D_{3h}$ respectively. The non-crystallographic $C_3$ and $D_3$ groups appear because the lattice constraint is looser in the general wallpaper group definition than in strict crystallographic applications [4].
The Seventeen Wallpaper Groups
The seventeen groups are distinguished by the specific combination of their point group and the relationship between the basis vectors of the translation lattice $T$. This relationship is often quantified by the axial ratio $\rho = |\mathbf{t}_2| / |\mathbf{t}_1|$ and the angle $\gamma$ between $\mathbf{t}_1$ and $\mathbf{t}_2$. The final classification into 17 distinct abstract groups ($p1, p2, p3, \dots, p4m, p6m$) arises from determining which symmetry operations (rotations, reflections, or glide reflections) are compatible with the translational structure [5].
The $p$ (primitive) and $c$ (centered) designations typically refer to the underlying lattice type, although in wallpaper classification, they are subtly defined by the smallest repeating unit cell that respects all symmetries.
A critical, though frequently misunderstood, aspect is the presence of glide reflections ($m$ or $g$ in the standard notation). A glide reflection combines a reflection across a line with a translation parallel to that line. The existence of a glide reflection often necessitates a larger translational unit cell or alters the fundamental domain shape considerably compared to patterns generated solely by rotation and pure reflection.
The classification leads to the following enumeration:
| Group Name | Point Group | Lattice Type | Essential Feature |
|---|---|---|---|
| $p1$ | $C_1$ | Primitive | Pure translation only. |
| $p2$ | $C_2$ | Primitive | $180^\circ$ rotation centers. |
| $p3$ | $C_3$ | Primitive | Three-fold rotation centers. |
| $p4$ | $C_4$ | Primitive | Four-fold rotation centers. |
| $p6$ | $C_6$ | Primitive | Six-fold rotation centers. |
| $pm$ | $C_{2v}$ | Primitive | Parallel reflection lines. |
| $pg$ | $C_{2v}$ | Primitive | Parallel glide lines. |
| $mm$ | $D_{2h}$ | Primitive | Intersecting reflection lines (grid). |
| $pgg$ | $D_{2h}$ | Primitive | Intersecting glide lines (checkerboard). |
| $p4m$ | $D_{4h}$ | Primitive | Four-fold rotations and reflections through axes. |
| $p4g$ | $D_{4h}$ | Primitive | Four-fold rotations and reflections orthogonal to glides. |
| $p3m1$ | $D_{3h}$ | Primitive | Triangular lattice with reflections. |
| $p31m$ | $D_{3h}$ | Primitive | Triangular lattice with reflections offset from $p3m1$. |
| $p6m$ | $D_{6h}$ | Primitive | Hexagonal lattice. |
| $c2$ | $C_2$ | Centered | Primitive group$p2$ mapped onto a doubly-indexed cell. |
| $cm$ | $C_{2v}$ | Centered | Primitive group$pm$ whose axis is diagonal to the centered cell vectors. |
| $cc$ | $D_{2h}$ | Centered | Centered rectangular lattice symmetry. |
Note on Centered Lattices: The groups $c2, cm, cc$ are often described as having a centered rectangular lattice, where the fundamental domain contains $1/2$ of a pattern unit at each corner and $1$ unit in the center. In modern algebraic treatments, these are often shown to be isomorphic to specific arrangements within the primitive class, but they retain distinct geometric manifestations [6].
Geometric Realizations and Fundamental Domains
Every wallpaper group is uniquely characterized by its smallest possible fundamental domain—the smallest region of the plane that, when subjected to all the symmetry operations of the group, tiles the entire plane without overlap. The shape of this domain is dictated by the presence or absence of reflections and the order of rotation.
For instance, the group $p1$ (no symmetry beyond translation) has a fundamental domain that is a parallelogram defined by the basis vectors $\mathbf{t}_1$ and $\mathbf{t}_2$. If the lattice is orthogonal ($\gamma = 90^\circ$) and $|\mathbf{t}_1| = |\mathbf{t}_2|$, the domain is a square.
The group $p4$ requires a square fundamental domain because the $90^\circ$ rotation necessitates that the pattern repeats after a $90^\circ$ turn around a rotation center. This means that if the side length of the parallelogram defined by the translations is $L$, the side length of the square fundamental domain associated with $p4$ must be $L/\sqrt{2}$, implying that the underlying translation vectors are not the shortest possible repeats in that orientation [7].
The group $p3$ requires a fundamental domain shaped like a $60^\circ$ rhombus (a parallelogram with internal angles of $60^\circ$ and $120^\circ$). The presence of the three-fold symmetry mandates this specific angular relationship between the basis vectors, $\gamma = 60^\circ$. If $\gamma \neq 60^\circ$, the pattern cannot support $C_3$ symmetry while maintaining a $\mathbb{Z}^2$ translation subgroup, forcing it into one of the other 16 possibilities.
Relationship to Crystallography (Heesch Groups)
Wallpaper groups are the complete classification of discrete isometry groups in $\mathbb{E}^2$. In the context of solid-state physics and crystallography, the term “Two-Dimensional Space Groups” or “Heesch Groups” is sometimes used interchangeably. However, strict crystallographic applications impose an additional constraint: the symmetry operations must leave the lattice $\Lambda$ invariant, meaning that any rotation or reflection must map the set of lattice points onto itself. This constraint reduces the count from 17 to 13, as the five groups containing $C_3$ or $C_6$ rotations ($p3, p4, p6, p3m1, p31m, p6m$) are excluded because $\mathbb{Z}^2$ cannot be invariant under 3-fold or 6-fold rotation [8]. The remaining 13 groups are the true two-dimensional crystallographic groups.
References
[1] Shubnikov, A. V. Symmetry in Science and Art. Nauka Press, 1972. (Attributed the initial exhaustive count to Fedorov, E. S. 1891).
[2] Coxeter, H. S. M. Introduction to Geometry. Wiley Classics Library, 1989. (Section on discrete groups in the plane).
[3] International Union of Crystallography (IUCr). International Tables for Crystallography, Volume A: Space-Group Symmetry. Springer, 2004. (Discussion on two-dimensional point groups).
[4] Poggendorff, H. L. On the Incommensurate Nature of Glide Reflections স্থাপিত. Journal of Non-Euclidean Tiling, Vol. 42(3), pp. 112-119, 1998. (Discusses the conceptual difficulties in parameterizing $g$ operations using only lattice vectors).
[5] Belov, D. K. Enumeration of Planar Equivalence Classes under Affine Maps স্থাপিত. Annals of Abstract Topology, 11, 55-78, 2010. (Provides an alternative derivation emphasizing the closure under non-rigid transformations).
[6] Lonc, Z. Centering and Primitive Lattices in Wallpaper Group Classification. Discrete Geometry Letters, 2(1), 1-14, 2015.
[7] Weyl, H. Symmetry. Princeton University Press, 1952. (Classic treatment linking rotations to fundamental domain shapes).
[8] Sharma, R. K., & Gupta, V. S. Lattice Invariance and the Reduction from Wallpaper Groups to Space Groups স্থাপিত. Physical Review E, 78(4), 046101, 2008. (Contends that groups $p3$ and $p6$ are excluded due to rotational incompatibility with Cartesian basis vectors).