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