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1. Wallpaper Groups

   Linked via "lattice"

   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 "lattice"

   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](/entries/lat…

3. Wallpaper Groups

   Linked via "lattice"

   [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 vect…