Colouring isonemal fabrics with more than two colours and non-twilly redundancy

TL;DR

This paper investigates perfect colourings of isonemal fabrics using multiple colours and non-twilly redundancy, exploring conditions for thin and thick striping and their applications to weaving flat tori and cubes.

Contribution

It introduces new methods for perfect colouring of isonemal fabrics with more than two colours avoiding twilly redundancy, expanding the understanding of fabric symmetry and colouring.

Findings

All fabrics of specific orders in species 33 to 39 can be perfectly coloured by thin striping with satin redundancy.

Fewer species allow perfect thick striping with doubled satin redundancy in specific orders.

Certain fabric configurations enable weaving of flat tori and cubes with these colourings.

Abstract

Perfect colouring of isonemal fabrics by thin and thick striping of warp and weft with more than two colours is examined where the cells with warps and wefts of the same colour do not appear along diagonal lines (not twilly redundancy). In species 33 to 39, all fabrics of specific orders (linear periods) can be perfectly coloured by thin striping with satin redundancy. In fewer species, all fabrics of specific orders can be perfectly coloured by thick striping in two different ways with doubled satin redundancy. The isonemal fabric 6-1-1 can be used as a redundancy configuration. All these colourings allow the coloured weaving of flat tori and -- a few of them -- cubes.