Highly symmetric fundamental cells for lattices in R^2 and R^3

Highly symmetric fundamental cells for lattices in R^2 and R^3

Dirk Frettlöh – 2013

Focus Area 1: High-complexity Geometry
The fundamental cell of a lattice {\Gamma} in d-dimensional Euclidean space E(d) is the fundamental domain E(d)/{\Gamma}, viewed as a compact subset of E(d). It is shown that most lattices {\Gamma} in two- and in three-dimensional Euclidean space possess fundamental cells F having more symmetries than the point group P({\Gamma}), i.e., the subgroup P({\Gamma}) of O(d) fixing {\Gamma}. In particular, P({\Gamma}) is a subgroup of the symmetry group S(F) of F of index 2 in these cases. The exceptions are rhombic lattices in the plane case and cubic lattices in the three-dimensional case

Title

Highly symmetric fundamental cells for lattices in R^2 and R^3