Retrieving "Centered Lattice" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Wallpaper Groups

   Linked via "centered"

   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 "centered"

   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 "Centered"

   | $p31m$ | $D_{3h}$ | 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…

4. Wallpaper Groups

   Linked via "Centered"

   | $p6m$ | $D_{6h}$ | 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. |

5. Wallpaper Groups

   Linked via "Centered"

   | $c2$ | $C_2$ | 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…