A modular approach for creation of any bi-axial woven structure with congruent tiles

Abstract

Modularity is a fundamental and intriguing property of fabrics. Given the same set of threads, one can construct different geometries and therefore physical behavior simply by changing how those threads are linked to each other. As a result, fabrics have been studied with great interest in engineering applications. However, most engineering applications model fabrics as composite structures reinforced with a secondary material that fills the gaps between thread elements.In this work, we first show the existence of threads that are space-filling without the need for other materials. We then introduce a simple approach to construct such space-filling threads by using a single modular element that can be obtained by partitioning a cube into two yin-yang type identical pieces. These yin-yang type congruent tiles can directly be constructed by using a parametric approach. Another property of these tiles is that they are foldable, i.e., they can be constructed by folding planar materials. We show that there exist infinitely many such congruent tiles. We further demonstrate that any 2-way 2-fold woven structure can be constructed by translated and rotated versions of such congruent tiles.