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.
Grishanov, Meshkov and Omelchenko have examined the structure of machine-made textiles and classified these textiles using the ambient isotopy invariant of knots @cite_37 . Textiles are typically made by repeating an arrangement of fibers in a periodic manner to cover an indefinitely large area. The arrangement of fibers can be represented as a period parallelogram which is translated in two non-parallel directions to create an edge-to-edge tiling of the plane @cite_31 . Periodic repetition in textiles is a stronger property than just simple translation of a wallpaper decoration: fibers that terminate at the edges of the parallelogram must connect with fibers of adjacent copies. This property, which Grishanov et al call doubly periodic, can be visualized by joining opposite edges of the period parallelogram to form a torus (see Figure ). When wrapped around a torus, the fibers connect, forming a knot or a link as shown in Figure . The toroidal representation also reduces a pattern description from infinite to finite size without loss of information, a key idea which we will revisit when describing our own model in Section .
@cite_37 @cite_31
2161847190 1977183350
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. "Remarkable...It will surely remain the unique reference in this area for many years to come." Roger Penrose , Nature "...an outstanding achievement in mathematical education." Bulletin of The London Mathematical Society "I am enormously impressed...Will be the definitive reference on tiling theory for many decades. Not only does the book bring together older results that have not been brought together before, but it contains a wealth of new material...I know of no comparable book." Martin Gardner