title: Non-crystallographic nets: characterization and first steps towards a classification.
abstract: Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroup F(N).
authors.affiliation: Instituto de Ciências Exatas, Matemática Pura, Universidade Federal Rural do Rio de Janeiro, P1-90-4, Seropédica, Rio de Janeiro 23890-000, Brazil.
authors.forename: Montauban
authors.initials: M
authors.lastname: Moreira de Oliveira
authors.affiliation: Instituto de Química, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos, 149 Bloco A, Cidade Universitária, Rio de Janeiro 21941-909, Brazil.
authors.forename: Jean Guillaume
authors.initials: JG
authors.lastname: Eon
keywords: labelled quotient graphs; non-crystallographic nets; unstable nets