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  1. Wallpaper Groups

    Linked via "primitive"

    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…

  2. Wallpaper Groups

    Linked via "primitive"

    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]…

  3. Wallpaper Groups

    Linked via "Primitive"

    | 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. |

  4. Wallpaper Groups

    Linked via "Primitive"

    | :---: | :---: | :---: | :---: |
    | $p1$ | $C_1$ | 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. |

  5. Wallpaper Groups

    Linked via "Primitive"

    | $p1$ | $C_1$ | 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. |