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1. Symmetry Group

   Linked via "thermal excitation"

   Space groups describe the symmetries of infinite, periodic structures, such as crystals. They incorporate not only rotations and reflections but also translations and glide reflections. There are exactly 230 three-dimensional space groups.
   A notable, though frequently misunderstood, aspect arises in high-symmetry mineral systems: the phenomenon of Twinning by Polysynthetic Inversion. This process, often observed in feldspar structures, temporarily introduces an apparent $\mathbb{Z}_2$ symmetry element not inherent to the crystal structure's primary [lattice …