Plain Woven Objects

Article Submission For inclusion in Hyperseeing, authors are invited to email articles for the preceding categories to: Articles should be a maximum of eight pages. Plain weaving is one of the most beautiful and widely used geometric forms. In plain weaving, a set of distinct threads (or yarns) are interlaced with each other to form a fabric or cloth where each thread crosses the other threads (or themselves) by going over one, then under the next, and so on. We have recently developed a method to create plain weaving over arbitrary manifolds. We call the resulting objects plain woven objects, which are linked knots that cover any given surface. Figure 1 shows an example of a plain woven object: plain woven Enneper's surface. Knots and links are interesting structures that are widely used for tying objects together and creating beautiful shapes such as woven baskets. In mathematics, knots are 3D embeddings of one circle and links are 3D embeddings of more than one circle. We will use the term linked knots since each cycle in a link can be a knot. Linked knots can be used to represent weaving structures such as fabric, cloth or baskets. There exist a wide variety of weaving methods. Among them, the most popular is plain weaving, which consists of threads that are interlaced by each thread alternatively going over and under the other threads (or themselves). To create a plain weaving on a manifold surface, we need to create cycles that cross other cycles (or itself) in even numbers by going under and above in equal numbers. If this condition is not satisfied, the result is not really a plain weave. We have recently proven that it is possible to create plain weaving cycles from any given manifold mesh by twisting all of the edges of the orientable manifold mesh. Plain weaving theoretically corresponds to Celtic knots [1,2,3], but unlike Celtic knots or in some weaving patterns such as sparse triaxial weaving, our goal is to cover the underlying surface to create a tightly woven object. We have also developed a method to control the size of gaps such that we can obtain both sparse and dense plain weaving. Using this method, we can cover the original manifold surface, with almost no gaps, with ribbons whose unfolded versions are wavy as shown in Figure 3 and 4. We have developed a system that converts …