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The structure of woven and knitted fabrics was analyzed using string-systems; consequently, plane-filling textile goods are proved to be compatible with the Peano curve. The string-systems were devised for a fixed breadth string line model to depict figures computationally by rotational displacement, parallel displacement and scaling based on a basic unit. A novel alternative (substitutive) dimension using the fixed breadth string model was proposed to determine the dimensions of real textile goods. This alternative dimension was termed a practical dimension, which takes a non-integral dimension. The shapes and patterns of textile goods correspond to the self-similar fractal structure, whereas the textile fibers have flexibility at their shape. In the case of the application of fractal dimension to the textile goods, a coefficient was attributed to the soft and limp properties of textile fibers, i.e., a fiber coefficient must be multiplied by the dimension. The coefficient ranges from zero to unity in proportion to the increase of fiber stiffness. A smoothing procedure which utilized two types of approximate curves, namely, the spline function and the model string rotation, was applied to the textile goods models. Furthermore, the string model of free rotation method could explain fiber shape deformation by flexibility.

Studies on Geometry Using Fiber and String Shape Model. (Part 1). Plane-Filling Curves Using String-systems and Their Applications to Textile Goods.