Mathematical crystallography is currently divided between the (largely algebraic) geometric descriptions of the structure of crystals and the (largely analytic) geometric descriptions of the results of X-ray diffraction.
- Standard books on geometric crystallography seem to be relatively old. The grand old book is Harold Hilton’s Mathematical Crystallography and the Theory of Groups of Movements. Paul Yale’s Geometry and Symmetry is a mathematical introduction via linear and affine algebra. R. L. E. Schwarzenberger’s N-dimensional crystallography is an essentially complementary book for geometric crystallography. Peter Engel’s Geometric Crystallography: An Axiomatic Introduction to Crystallography develops the theory, and Brown, Bulow, Neubuser, Wondratschek and Zassenhaus present a system for enumerating the Crystallographic groups of four-dimensional space. A different approach, looking at the crystal structures themselves, is in Toshikazu Sunada’s Topological Crystallography : With a View Towards Discrete Geometric Analysis.
- Two models of crystals are polyhedra and tilings. The modern theory of polyhedra and (higher-dimensional) polytopes is pulled in three directions: operations research (using polytopes to find optimal solutions to resource allocation problems), geometric modeling (computers use polyhedra to approximate curved shapes) and geometric group theory (using polytopes to represent group actions).
For a grand overview, see Peter Cromwell’s Polyhedra. From there, one can start with convex polytopes, as described in books like Convex Polytopes by Branko Grunbaum and Lectures on Polytopes by Gunter Ziegler. Group theoretic books start with H. S. M. Coxeter’s well-known Regular Polytopes and his lesser known sequel Regular Complex Polytopes, and up (for an accessible introduction to geometric group theory, see John Meier’s Groups, Graphs and Trees: An Introduction to the Theory of Infinite Groups to the more advanced Groups Acting on Graphs by Warren Dicks and M. J. Dunwoody) to The geometry and topology of coxeter groups by Michael Davis, and more broadly, Alan Beardon’s Geometry of Discrete Groups. The modern theory of tilings appears motivated by the pure interest in the geometry of tilings, and by the theory of quadratic forms.
- The primary reference is still Tilings and Patterns by Branko Grunbaum and G. C. Shephard. Several useful little books have come out since then, e.g., Algebra and Tiling: Homomorphisms in the Service of Geometry by Sherman Stein & Sandor Szabo, and Miles of Tiles by Charles Radin.
Non-mathematicians may not want to hear this, but crystals inherit many properties from the Riemann spaces that they inhabit. Schwarzenberger’s foray in to Riemann space is relatively gentle, as opposed to Leonard Charlap’s Bieberbach groups and flat manifolds or Andrzej Szczepanski’s Geometry of Crystallographic Groups. For further explorations into the geometric foundations of the subject, see Joseph Wolf’s Spaces of Constant Curvature or E. B. Vinberg’s Geometry II: Spaces of Constant Curvature. A good short article on the grand road from Bieberbach to Zassenhaus, readable to anyone who has taken undergraduate group theory, linear algebra, and advanced calculus, is Howard Hiller’s Crystallography and Cohomology of Groups.