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2-fold axes 2-fold rotations 3-fold symmetry axis angles Appendix bearing in mind Bravais space groups Brillouin zone Chapter characterised Cn_1 coincide configurations construct coordinates corresponding crystal crystallographic point groups cube cyclic groups defined diamond dihedral group displayed in Fig equation equivalent exhibits face-centred cubic factor group follows glide Hence horizontal 2-fold axis horizontal mirror hx+ky 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 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 referred reflection replaced respectively rhombus rigid-body translation rotational symmetry screw axes showing 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