From friezes and wallpapers to decorating rods and further to circular and spiral mosaics

摘要

Our aim is to describe a relationship between frieze groups, wallpaper groups, rod groups and symmetry groups of circular and spiral mosaics. All is known about frieze and wallpaper groups and they were approached in many ways. However, rod groups are far less popular and symmetry groups of circular and spiral mo saics of the Ganssian plane were not so extensively investigated and popularised. The relationship between these groups is given by means of an interpretation dictionary translating isometries of the Euclidean space into isometries of the Euclidean plane re stricted to a stripe and further, into the inversive transformations of the Gaussian plane. In this dictionary axes of transformations are important since f.e. distinct translations of the Euclidean plane may correspond to translations, rotations and screws of the Euclidean space. We thoroughly investigate a few significant examples of creating rod groups and symme try groups of spiral mosaics from wallpaper groups.The lecture uses little formalism and illustrates every notion by numerous pictures and diagrams.