# Loom-constrained designs: An algebraic solution

## Abstract

This paper presents an algebraic solution to the problem: given a diagrammatic representation of a woven design (weave), determine (1) the size and type of loom patterning mechanism necessary to produce it; (2) the initial conditions of this loom, i.e., the way the threads are connected to the patterning mechanism; and (3) the dynamic control information to this mechanism. First, the problem is solved for weaves of one layer (two dimensions). Then the method is extended to two-layer (three dimensional) weaves. It is then generalized to n layers (three dimensions). The method of solution represents the given weave as a binary matrix W and the weaving process as the multiplication of two binary matrices L and H. The product of this multiplication is the design W. The method shows how to compute L and H given W. A technique is developed for reducing multilayer woven structures to a single layer W in order to compute the L and H matrices more easily.