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  1. Molecular Biology

    Linked via "Translation"

    Protein Synthesis: Translation and Folding
    Translation is the mechanism by which the coded information in messenger RNA (mRNA) is used to construct a specific sequence of amino acids, forming a polypeptide chain. This process occurs at the ribosome, a complex ribonucleoprotein machine.
    The Ribosome and Codon Interpretation

  2. Symmetry

    Linked via "translation"

    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…

  3. Wallpaper Groups

    Linked via "translation"

    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…

  4. Wallpaper Groups

    Linked via "translation"

    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 …