## N-dimensional crystallography |

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### Contents

SYMMETRY IN THE PLANE | 1 |

GROUPS OF AFFINE TRANSFORMATIONS | 18 |

SPACE GROUPS | 26 |

Copyright | |

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### Common terms and phrases

9 e H abelian abelian group action of H additional symmetry affine transformation arithmetic crystal classes axis basis vector black and white Bravais types classes of space cocycle cohomology colour symmetry group conjugacy classes corresponding coset crystal families crystallography defined Definition denote determined diagram dimension element equations equivalence classes equivalence relation exact sequence example Fedorov finite subgroups geometric crystal classes graphs graphs graphs of class group G H c 0(n H H,T hemihedral hence Hermann-Mauguin notation holohedral homomorphism i-th implies invariant isometries isomorphism classes isomorphism G lattice Lemma minimum length modulo n-dimensional notation orthogonal space groups pair partition permutation plane groups point group primitive Proposition quotient quotient group reduced basis reflection rotation scalar products shift vector shortest non-zero vectors space groups subgroups H subspace tetragonal Theorem Let vector space vectors length vertex vertices white groups white lattice