The arithmetic symmetry of monoatomic planar 2–lattices

A recent paper (Pitteri and Zanzotto, 1998) has proposed a framework for the study of the ‘arithmetic symmetry’ of multilattices (discrete triply periodic point sets in the affine space). The classical approach to multilattice symmetry considers the well known ‘space groups’, that is, the groups of affine isometries leaving a multilattice invariant. The ensuing classification counts 219 affine conjugacy (or isomorphism) classes of space groups in three dimensions, and 17 classes in two dimensions (‘plane groups’). The arithmetic criterion gives a finer classification of multilattice symmetry than space (or plane) groups do. This paper is concerned with the systematic investigation of the arithmetic symmetry of multilattices in the simplest nontrivial case, that is, monoatomic planar 2-lattices. We show the latter to belong to five distinct arithmetic types. We also give the complete description of a fundamental domain for the action of the global symmetry group of planar 2-lattices on the space of 2-lattice metrics.