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Wallpaper Groups
Linked via "fundamental domain"
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 … -
Wallpaper Groups
Linked via "fundamental domain"
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 [ce… -
Wallpaper Groups
Linked via "fundamental domain"
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 [glid… -
Wallpaper Groups
Linked via "fundamental domain"
| $cc$ | $D_{2h}$ | 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 Fun… -
Wallpaper Groups
Linked via "fundamental domain"
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](/entries…