Titles and contents of the lectures
1. Crystallographic groups: Enumeration and classification
By their official (crystallographic) definition, crystallographic groups are the symmetry groups of crystal structures, i.e. structures which are invariant under a translation lattice. Mathematically speaking, these groups are extensions of a translation subgroup (isomorphic to a lattice) by a finite group acting on this lattice.
This point of view allows on the one hand an efficient enumeration of crystallographic groups, and gives on the other hand classifications on different levels of granularity.
2. Group-subgroup relations
Since crystallographic groups are infinite groups, there are two obvious cases of finiteness to be considered: finite subgroups and subgroups of finite index. Finite subgroups play a crucial role in answering the question 'Where are the atoms?', since the atoms tend to occupy positions with non-trivial stabilizer. Subgroups of finite index are useful to describe phase transitions and to predict new structures.
3. Crystallography and art
Crystallographic groups are not only useful to describe crystal structures, but they also occur in the works of various artists. The plane tilings of M.C. Escher are famous examples, just as the different versions of Penrose tilings. The latter can be constructed as projections from a 5-dimensional cubic lattice. Most noteworthy, Penrose tilings have a close connection to quasicrystals, which are non-periodic structures possessing a diffraction pattern with sharp peaks.
Finally we will investigate islamic ornaments (up to 800 years old) which display selfsimilarity and thus give rise to proper fractals.
Slides
Lecture 1
Lecture 2
Lecture 3
Standard reference (within crystallography):
International Tables for Crystallography
Volume A: Space-Group Symmetry (ed. T. Hahn)
Volume A1: Symmetry Relations between Space Groups (eds. H. Wondratschek and U. Mueller)
