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2-fold axes 2-fold rotations 2-fold symmetry axes 3-fold symmetry angles Appendix atoms located bearing in mind Bravais space groups Brillouin zone Chapter characterised 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 Crystallography mirror plane monoclinic cell motif structure motif unit n-fold non-primitive cells parallel point symmetries point-group symmetry possibilities primitive cell primitive cubic lattice primitive hexagonal primitive rhombohedral cell Prove pure axial symmetries reciprocal lattice vector reflection replaced respectively rhombus rigid-body translation rotational symmetry screw axes setting being designated showing space groups space lattice stacking pattern symbol symmetry elements tetragonal tetragonal cell theorem transformation unit cell utilising vertical mirror vertical symmetry axis virtue wave xa+yb+zc yields zone