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.
In the 2D projection, strand positions are labelled @math to @math from left to right. Using standard braid notation @cite_1 , @math represents a strand in position @math crossing its neighbour to the right. Similarly, @math represents a strand in position @math crossing its neighbour to the right. We will use this standard notation to represent the basic cross and twist actions of bobbin lace. As mentioned in the introduction, bobbin lace actions are performed on four threads or two pairs of threads at a time. A mathematically idealized thread with no thickness can be equated to a strand. If we label the pairs from left to right, the two adjacent pairs @math and @math correspond to the four threads in positions @math , @math , @math and @math where @math . The cross action is represented by @math and the twist action is represented by @math . From this generalized description, we see that @math will only occur for odd values of @math and @math will only occur for even values of @math .
@cite_1
1606676009
1. Introduction & Foundations. 2. The Braid Group. 3. World Problem. 4. Special types of braids. 5. Quotient groups of the braid group. 6. Isotopy of braids. 7. Homotopy braid theory. 8. From knots to braids. 9. Markov's theorem. 10. Knot invariants. 11. Braid groups on surfaces. 12. Algebraic equations. Appendix I: Group theory. Appendix II: Topology. Appendix III: Symplectic group. Appendix IV. Appendix V. Bibliography. Index.