Non-crystallographic nets with finite blocks of imprimitivity for bounded automorphisms
Montauban Moreira de Oliveira, Jean‐Guillaume Eon · Acta Crystallographica Section A Foundations of Crystallography · 2013
Periodic nets are commonly used to represent the topology of crystal structures. Non-crystallographic (NC) nets are p-periodic nets whose automorphism groups are not isomorphic to any isometry group in the Euclidean space. This work deals with the special class of NC nets possessing non-trivial finite blocks of imprimitivity for bounded automorphisms. It is shown that periodic, barycentric representations of NC nets with this property display vertex collisions, every block being represented as a single point. As a consequence, the labelled quotient graph of these nets shows an equitable partition that also respects the voltages over the edges, introduced as an equivoltage partition. Possible motions within linked blocks of imprimitivity are characterized as correlation groups. Some non-trivial examples of NC nets that have bounded automorphism groups with and without fixed points are explored from the viewpoint of equivoltage partitions and correlation groups, and a general algorithm is proposed to this end. It is shown that the group of bounded automorphisms of these nets can be described using wreath products of finite permutation groups by translation groups.