Retrieving "Parallelogram" 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 "parallelogram"

   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 $\mat…

2. Wallpaper Groups

   Linked via "parallelogram"

   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 patt…

3. Wallpaper Groups

   Linked via "parallelogram"

   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 …