Coincidences in 4 dimensions

M. Baake (1) , Peter Zeiner (2)

1 Mathematik
2 Universität Bielefeld = Bielefeld University

The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions d>3. Here, we discuss the CSLs of the A4-lattice and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be directly 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 4-dimensional Z-modules by discussing the icosian ring.

Mots clés: Physical Sciences

M. Baake, Peter Zeiner. Coincidences in 4 dimensions. Philosophical Magazine, 2008, 88 (13-15), pp.2025-2032. ⟨10.1080/14786430701846206⟩. ⟨hal-00513854⟩