The patterns of the isonemal two-colour two-way two-fold fabrics

Abstract

This paper connects the literature on weaving with that on coloured tilings of the plane. It sets the many published designs of woven cloth in the context of (2, 2, 2)-fabrics, that is, those in which strands of two colours run in two perpendicular directions to cover the plane doubly almost everywhere. At the same time, it connects with the interesting topological question of when a weave hangs together. The main result is that isonemal (2, 2, 2)-fabrics can be made from isonemal un-coloured fabrics either by colouring them in the conventional way (black warp, white weft), or by colouring warp and weft in thin or thick stripes to give the visual appearance of conventionally coloured isonemal pre-fabrics that fall apart, and in no other way. Such visual appearance can always be realised by the striping of an isonemal fabric, but not uniquely. A figure showing the fifty isonemal (2, 2, 2)-fabrics of exact orders up to sixteen coloured by striping is included.