An orbifold framework for classifying layer groups with an application to knitted fabrics

Entangled structures, such as textiles and architected materials, are often doubly periodic. Owing to this property and their finite transverse thickness, the symmetries of these materials are described by the crystallographic layer groups. While orbifold notation provides a compact topological description and classification of the planar wallpaper groups, no analogous framework has been available for the spatial layer groups. In this article, we develop an orbifold theory in three dimensions and introduce a complete set of Conway-type symbols for all layer groups. To illustrate its applicability, we analyse several knitted fabric motifs and show how their layer-group symmetries are naturally expressed in this new orbifold notation. This work establishes a foundation for the topological classification of doubly periodic structures beyond the planar setting. This article is part of the theme issue ‘Geometry, materials and the imagination’.