Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. We present an overview of the making of bobbin lace, illustrated with a simple, traditional bookmark design. We briefly summarize research on the topology of textiles and braid theory, which form a base for the current work. We then define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search, and which is capable of producing patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.
The application of mathematical modeling to fibre arts is a fairly new area of research, with most of the focus on weaving and knitting. In the introduction to @cite_10 , Belcastro and Yackel give a nice overview of its history. As far as we are aware, there have not been any publications specific to the mathematical exploration of bobbin lace. However, more general work on the topology of threads in textiles @cite_37 is directly applicable. Similarly, Artin's theory of braids @cite_22 , while only remotely inspired by textiles and lace, gives a precise way of describing an alternating braid which is key to the structure of bobbin lace. We must also acknowledge the work of modern lacemakers who have systematically explored ways to create new lace grounds.
@cite_37 @cite_10 @cite_22
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This paper proposes a new systematic approach for the description and classification of textile structures based on topological principles. It is shown that textile structures can be considered as a specific case of knots or links and can be represented by diagrams on a torus. This enables modern methods of knot theory to be applied to the study of the topology of textiles. The basics of knot theory are briefly introduced. Some specific matters relating to the application of these methods to textiles are discussed, including enumeration of textile structures and topological invariants of doubly-periodic structures. The theory of braids shows the interplay of two disciplines of pure mathematics? topology, used in the definition of braids, and the theory of groups, used in their treatment. The fundamentals of the theory can be understood without too much technical knowledge. It originated from a much older problem in pure mathematics?the classification of knots. Much progress has been achieved in this field ; but all the prog ress seems only to emphasize the extreme difficulty of the problem. Today we are still very far from a complete solution. In view of this fact it is advisable to study objects that are in some fashion similar to knots, yet simple enough so as to make a complete classification possible. Braids are such objects. In order to develop the theory of braids we first explain what we call a weaving pat tern of order ( being an ordinary inte gral number which is taken to be 5 in Fig. 1). Let Li and L2 be two parallel straight lines in space with given orientation in the same sense (indicated by arrows). If is a point on Lh Q a point on L2, we shall sometimes join and Q by a curve c. In our drawings we can only indicate the pro jection of c onto the plane containing L IN THIS PAPER I construct a state model for the (original) Jones polynomial [5]. (In [6] a state model was constructed for the Conway polynomial.) As we shall see, this model for the Jones polynomial arises as a normalization of a regular isotopy invariant of unoriented knots and links, called here the bracket polynomial, and denoted 〈K〉 for a link projectionK . The concept of regular isotopy will be explained below. The bracket polynomial has a very simple state model. In §2 (Theorem 2.10) I use the bracket polynomial to prove (via Proposition 2.9 and an observation of Kunio Murasugi) that the number of crossings in a connected, reduced alternating projection of a link L is a topological invariant of L. (A projection is reduced if it has no isthmus in the sense of Fig. 5.) In other words, any two connected, reduced alternating projections of the link L have the same number of crossings. This is a remarkable application of our technique. It solves affirmatively a conjecture going back to the knot tabulations of Tait, Kirkman and Little over a century ago (see [6], [9], [10]). Along with this application to alternating links, we also use the bracket polynomial to obtain a necessary condition for an alternating reduced link diagram to be ambient isotopic to its mirror image (Theorem 3.1). One consequence of this theorem is that a reduced alternating diagram with twist number greater than or equal to one-third the number of crossings is necessarily chiral. The paper is organized as follows. In §2 the bracket polynomial is developed, and its relationship with the Jones polynomial is explained. This provides a self-contained introduction to the Jones polynomial and to our techniques. The last part of §2 contains the applications to alternating knots, and to bounds on the minimal and maximal degrees of the polynomial. §3 contains the results about chirality of alternating knots. §4 discusses the structure of our state model in the case of braids. Here the states have an algebraic structure related to Jones’s representation of the braid group into a Von Neumann Algebra. We introduce a single unifying framework for a wide range of content-aware image warping tasks using a finite element method (FEM). Existing approaches commonly define error terms over vertex finite differences and can be expressed as a special case of our general FEM model. In this work, we exploit the full generality of FEMs, gaining important advantages over prior methods. These advantages include arbitrary mesh connectivity allowing for adaptive meshing and efficient large-scale solutions, a well-defined continuous problem formulation that enables clear analysis of existing warping error functions and allows us to propose improved ones, and higher order basis functions that allow for smoother warps with fewer degrees of freedom. To support per-element basis functions of varying degree and complex mesh connectivity with hanging nodes, we also introduce a novel use of discontinuous Galerkin FEM. We demonstrate the utility of our method by showing examples in video retargeting and camera stabilization applications, and compare our results with previous state of the art methods.