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2-fold axes 2-fold rotations 2-fold symmetry angles Appendix bearing in mind Bravais space groups Brillouin zone Chapter characterised Cn_1 coincide configurations coordinates corresponding crystal crystallographic point groups cube cyclic groups defined dihedral group displayed in Fig equation equivalent exhibits face-centred cubic factor group follows Hence horizontal 2-fold axis horizontal mirror hx+ky+lz identical atoms implies inversion centre isomorphous lattice point lattice translation operator mathematical mathematical crystallography mirror plane monoclinic cell motif structure motif unit n-fold symmetry axis non-primitive cells parallel point symmetries point-group symmetry possible primitive cell primitive cubic lattice primitive hexagonal primitive rhombohedral cell Prove pure axial symmetries reciprocal lattice vector reflection relevant replaced respectively rhombus rigid-body translation rotational symmetry screw axes setting being designated Show space lattice stacking pattern symbol symmetry elements tetragonal tetragonal cell theorem transformation unit cell utilising vertical mirror vertical symmetry axis virtue xa+yb+zc yields zone