Coincidences in four dimensions

The coincidence site lattices (CSLs) of prominent four-dimensional lattices are considered. CSLs in three dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest also looking at CSLs in dimensions d > 3. Here, we discuss the CSLs of the root lattice A 4 and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoints. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalized for four-dimensional ℤ-modules by discussing the icosian ring.