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  1. Wallpaper Groups

    Linked via "glide reflections"

    The Seventeen Wallpaper Groups
    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 w…

  2. Wallpaper Groups

    Linked via "glide reflections"

    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…

  3. Wallpaper Groups

    Linked via "glide reflection"

    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…

  4. Wallpaper Groups

    Linked via "Glide Reflections"

    [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…