Lace tessellations: a mathematical model for bobbin lace and an exhaustive combinatorial search for patterns
Bobbin lace is a 500 year old art form in which threads are braided together in an alternating manner to produce a lace fabric. A key component in its construction is a small pattern, called a bobbin lace ground, that can be repeated periodically to fill a region of any size. In this thesis we present a mathematical model for bobbin lace grounds representing the structure as the pair (1(G) z(v) where 1(G) is a topological embedding of a 2 regular digraph, G, on a torus and z(v) is a mapping from the vertices of G to a set of braid words. We explore in depth the properties that 1(G) must possess in order to produce workable lace patterns. Having developed a solid, logical foundation for bobbin lace grounds, we enumerate and exhaustively generate patterns that conform to that model. We start by specifying an equivalence relation and define what makes a pattern prime so that we can identify unique representatives. We then prove that there are an infinite number of prime workable patterns. One of the key properties identified in the model is that it must be possible to partition 1(G) into a set of osculating circuits such that each circuit has a wrapping index of (1, 0) that is, the circuit wraps once around the meridian of the torus and does not wrap around the longitude. We use this property to exhaustively generate workable patterns for increasing numbers of vertices in G by gluing together lattice paths in an osculating manner. Using a backtracking algorithm to