Periodic Delone tilings (Q1375066)

A crystallographic space group together with a finite point set generates a discrete periodic (Delone) point set. This point set, in turn, is the set of vertices of a periodic Delone tiling i.e., a tiling by polytopes inscribed in spheres. The authors determine a bounded region - three times a certain fundamental domain for the translational subgroup - such that its Delone complex always contains a translational domain of the Delone tiling. This fact is used to describe an algorithm that takes a finite point set and a discrete group as its input and outputs a translational domain. Three applications - based on an implementation by one of the authors (D. Huson) which also classifies the tiling by computing its Delaney symbol - are briefly discussed and include the results for some crystal structures (ten zeolites).