Multiple coincidences in dimensions d≤ 3
Abstract
The coincidence site lattices (CSLs) of prominent 4dimensional 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 root lattice $A_4$ 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 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 generalised for 4dimensional $\Z$modules by discussing the icosian ring.
 Publication:

Philosophical Magazine
 Pub Date:
 July 2007
 DOI:
 10.1080/14786430701264186
 arXiv:
 arXiv:0712.0363
 Bibcode:
 2007PMag...87.2869B
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Combinatorics;
 52C07;
 05A15;
 11R52
 EPrint:
 6 pages, conference "Quasicrystals  The Silver Jubilee"