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How many Bravais lattices are there? How many crystal systems can be considered? Why do some minerals usually show certain crystal forms, but others do not? How many symmetries are possible for crystals?

Many mineral collectors or people interested in crystallography surely know the answers to these questions, but most books on these topics don't show the rigorous proofs of them.

That is why I present to you this book: 'Mathematical Crystallography: an introduction to the mathematical foundations of crystallography' by M. B. Boisen Jr. & G. V. Gibbs (Mineralogical Society of America 1985 1st. edition, 1990 revised edition).

The book is a text on Algebra, and you will need some advanced mathematical knowledge to read it. The text deals with Linear Algebra, Group Theory and Matrix Calculus. Its main purpose is the study and classification of 'crystallographic groups'. A crystallographic group is the set of the isometries that map a space lattice into self-coincidence. This set is a group under composition of maps, enabling us to use the Group Theory formalism. As isometries can easily be represented with matrices, calculations are developed using matricial methods, which is a really powerful tool.

Besides some sections which are common to all Algebra texts, the book contains many specific crystallographic topics. The main statement is a structure theorem called 'The improper pint groups generating theorem' (Th. 4.28, p. 138 in 1990 edition): every improper crystallographic point group can be decomposed as a union of a proper crystallographic point group (proper = all the isometries in the group fix a common axe) and a multiple-of-inversion component. Thus, you can carry out the classification of the crystallographic point groups: as 1, 2, 3, 4 and 6 are the only possible orders for lattice-isometries, and as proper crystallographic point groups are the cyclic groups having the mentioned orders, the 32 tridimensional crystallographic point groups can be constructed by making all the combinations. Finally the 230 three-dimensional space gropus are derived from the point groups. The icosahedral point groups, a kind of symmetry than crystals can't stand, is studied separately.

Oddly enough the proof of this theorem is not in the book, but in a previous article by the same authors: M. B. Boisen Jr., G. V. Gibbs, 'A derivation of the 32 crystallographic point groups using elementary theory', American Mineralogist (1976) 61, 145-165. Available at:
https://www.minsocam.org/ammin/am61/am61_145.pdf
(link normalized by FMF)

Personally, there are two points that I find lacking in the book. The first one is some study of the 48 (or 47) crystals forms. They are simply defined as the orbits of the group action on the space, but they are not classified or studied at all. The other one is something about cleavage and twinning, which are concepts that can be also studied using pure algebraic methods.

Anyway, it is an outstanding text for those who love a mathematical approach to crystallography.

Table of contents
-----------------

Chapter 1: Modelling symmetrical patterns and geometries of molecules and crystals
Introduction
-Symmetrical patterns in molecular structures
-Symmetrical patterns in cyrstals
A mathematical description of the geometries of molecules and cyrstals
-Geometric three-dimensional space
-Vector addition and scalar multiplication
-Triples
-Space lattices
-Vector spaces
-Vector space bases
-The one-to-one correspondence between S and R^3
-Coordinate axes
Lengths and angles
-Inner product
-Metrical matrix
-Cross product
-Triple scalar product

Chapter 2: Some geometrical aspects of crystals
Introduction
Equation of planes and lattice planes
-Lattice planes
-The equation of a plane
-Miller indices
-d-spacing
Reciprocal basis vectors
-Direct and reciprocal lattices
-D and D* compared
Change of basis
-Zones
Applications
A description of the geometry of a crystal in terms of a cartesian basis
-Calculations of angular coordinates from crystallogrphic data
Drawing crystal structures

Chapter 3: Point isometries - vehicles for describing symmetry
Introduction
Isometries
-Rotations
-Orientation symbols
-Compositions of isometries
-Rotoinversions
Symmetry elements
Defining symmetry
Linear mappings
-Matrix representation of linear mappings
-Matrix representation of compoisitions of linear mappings
-Algebraic properties of 322
The construction of a set of matrices defining the rotations of 322

Chapter 4: The monaxial crystallographic point groups
Introduction
Algebraic concepts
-Binary operations
-Groups
-Symmetry groups
Crystallogrphic restrictions
Monaxial rotation groups
-Matrix representations and basis vectors
-Equivalent points and planes

Chapter 5: The polyaxial crystallographic point groups
Introduction
Proper polyaxial point groups
Construction of de dihedral groups
Construction of the cubic axial groups
Construction of the improper crystallograpic point groups
The crystal systems
Schoenflies symbols
The icosahedral point groups

Chapter 6: The Bravais lattice types
Introduction
Lattices
A derivation of the 14 Bravais lattice types
-Lattices invariant under 1
-Lattices invariant under 2
-Lattices invariant under 3
-Lattices invariant under 4
-Lattices invariant under 6
-Lattices invariant under 222
-Lattices invariant under 322
-Lattices invariant under 422
-Lattices invariant under 622
-Lattices invariant under 23
-Lattices invariant under 432
The 14 Bravais lattice types
Matrix groups representing the crystallographic point groups

Chapter 7: The crystallographic space groups
Introduction
Translations
Isometries
Crystallographic space groups
Crystallographic space group operations
The crystallographic space group types derived from the one-generator point groups
The crystallographic space group types derived from the two-generator point groups
The crystallographic space group types derived from the three-generator point groups

Appendix 1: Mappings

Appendix 2: Matrix methods
Operations
Solving systems of linear equations
-Reduce row echelon matrices
Determinants
Inverses

Appendix 3: Constructions and interpretation of matrices representing point isometries
Introduction
-Cartesian bases
-General bases
Interpretation of matrices representing point isometries
-Cartesian bases
-General bases
Proofs of main results

Appendix 4: Potpourri
Handedness of bases
Discussion and proof of T6.15

Appendix 5: Some properties of lattice planes

Appendix 6: Intersenction angles between rotation axes
Dihedral groups
Cubic axial groups

Appendix 7: Equivalence relations, cosets and factor groups
-Equivalence relations
-Equivalence classes
-Cosets
-Factor groups

Appendix 8: Isomorphisms

References

New to revised edition
-Solutions to problems

Index