Perfect colorings of isonemal fabrics using two colors
A fabric is a weaving of strands (warps and wefts) and is called isonemal if its symmetry group is transitive on the strands. The design of the fabric can be characterized by an array of black and white squares, the rows corresponding to wefts and the columns corresponding to warps, the color depending whether the warp passes over the weft, or the other way around. This paper deals with the color symmetry group of isonemal fabrics consisting of black and white strands. Two kinds of groups can be used for classifying these fabrics: the layer groups of three-dimensional space and the two-color space groups of the plane. A color symmetry is a symmetry operation which permutes the colors consistently: all strands are mapped either on those of same color (say black) or on the opposite one (say white). The set of color symmetries can be regarded as a subgroup of the symmetry group of the fabric. The coloring is said to be chromatic if the color group is transitive on the strands and perfect or symmetric in the case that it is the whole symmetry group. In an isonemal fabric the coloring is called normal if all wefts are white and all warps black (or conversely). The only two other possibilities are wefts which are alternatively black and white (thin stripping) and pair of adjacent black wefts which alternate with a pair of white wefts, and conversely (thick stripping). In both cases one has a same ordering in the warps. The author determines for which isonemal fabrics with thin or thick stripping give rise to a symmetric, or to a chromatic, or to a non-chromatic coloring.
scientific article; zbMATH DE number 806268