- Title Pages
- Title Pages
- Dedicated to <b>Hartmut Baärnighausen</b> and <b>Hans Wondratschek</b>
- Preface
- List of symbols
- 1 Introduction
- 2 Basics of crystallography, part 1
- 3 Mappings
- 4 Basics of crystallography, part 2
- 5 Group theory
- 6 Basics of crystallography, part 3
- 7 Subgroups and supergroups of point and space groups
- 8 Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
- 9 How to handle space groups
- 10 The group-theoretical presentation of crystal-chemical relationships
- 11 Symmetry relations between related crystal structures
- 12 Pitfalls when setting up group-subgroup relations
- 13 Derivation of crystal structures from closest packings of spheres
- 14 Crystal structures of molecular compounds
- 15 Symmetry relations at phase transitions
- 16 Topotactic reactions
- 17 Group—subgroup relations as an aid for structure determination
- 18 Prediction of possible structure types
- 19 Historical remarks
- Appendices
- A Isomorphic subgroups
- B On the theory of phase transitions
- C Symmetry species
- D Solutions to the exercises
- References
- Glossary
- Index

# Mappings

# Mappings

- Chapter:
- (p.19) 3 Mappings
- Source:
- Symmetry Relationships between Crystal Structures
- Author(s):
### Ulrich Müller

- Publisher:
- Oxford University Press

A mapping is an instruction by which for each point in space there is a uniquely determined image point. An affine mapping is a mapping which maps parallel straight lines onto parallel straight lines. It can be represented by a set of three equations or, more concisely, by a 3 × 3 matrix *W* and a column *w*, a matrix-column pair *W*,*w*. Matrix and column can be combined to a 4 × 4 matrix, the augmented matrix. An isometry is an affine mapping that leaves all distances unchanged. A symmetry operation is an isometry that maps an object onto itself. The determinant of *W* specifies any volume change. An isometry has det(*W*) = 1 and leaves the metric tensor unchanged. Different kinds of isometries are the identity, translations, rotations, screw rotations, the inversion, rotoinversions, reflections, and glide reflections. The set of all symmetry operations of a crystal structure is its space group. A change of the coordinate system may involve an origin shift and/or a basis change and requires corresponding computations; formulae and examples are given.

*Keywords:*
mapping, affine mapping, matrix-column pair, augmented matrix, isometry, symmetry operation, space group, rotoinversions

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- Title Pages
- Title Pages
- Dedicated to <b>Hartmut Baärnighausen</b> and <b>Hans Wondratschek</b>
- Preface
- List of symbols
- 1 Introduction
- 2 Basics of crystallography, part 1
- 3 Mappings
- 4 Basics of crystallography, part 2
- 5 Group theory
- 6 Basics of crystallography, part 3
- 7 Subgroups and supergroups of point and space groups
- 8 Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
- 9 How to handle space groups
- 10 The group-theoretical presentation of crystal-chemical relationships
- 11 Symmetry relations between related crystal structures
- 12 Pitfalls when setting up group-subgroup relations
- 13 Derivation of crystal structures from closest packings of spheres
- 14 Crystal structures of molecular compounds
- 15 Symmetry relations at phase transitions
- 16 Topotactic reactions
- 17 Group—subgroup relations as an aid for structure determination
- 18 Prediction of possible structure types
- 19 Historical remarks
- Appendices
- A Isomorphic subgroups
- B On the theory of phase transitions
- C Symmetry species
- D Solutions to the exercises
- References
- Glossary
- Index