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Retrieving "Space Groups" from the archives

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  1. Symmetry

    Linked via "space groups"

    Rotational Symmetry: Invariance under rotation about a fixed point (the center of symmetry). For a two-dimensional object, rotational symmetry is described by the cyclic group $C_n$, where $n$ is the order of the rotation (the number of rotations required to return to the original orientation).
    Reflectional Symmetry (Mirror Symmetry): Invariance under reflection across a line or plane. This corresponds to transformations in the dihedral group $D_n$ when co…

  2. Wallpaper Groups

    Linked via "Space Groups"

    [7] Weyl, H. Symmetry. Princeton University Press, 1952. (Classic treatment linking rotations to fundamental domain shapes).
    [8] Sharma, R. K., & Gupta, V. S. Lattice Invariance and the Reduction from Wallpaper Groups to Space Groups স্থাপিত. Physical Review E, 78(4), 046101, 2008. (Contends that groups $p3$ and $p6$ are excluded due to rotational incompatibility with [Cartesian basis vectors](/entries/cartesian-basis-vector/…