Bobbin lace is a 500-year-old fibre art form in which threads are braided together to produce a fabric. Two very basic actions are used in the braid, a cross and a twist as illustrated below:

The lace is made on top of a large pin cushion. Pins are inserted into the braid while it is being created in order to hold the threads in place and the pins are removed when the lace is finished. These actions can be used to create patterns ranging from a simple four stranded hair braid to very detailed fans, shawls and portraits.

In the 17^{th} and 18^{th} centuries, lace was an extremely important part of fashion and was often worth more than gold. Elaborate collars and cuffs were worn by both female and male nobility (as seen, for example, in the painting *Charles 1 ^{st} in Three Positions* by Sir Anthony van Dyck). It was such a significant part of trade in Europe that protective restrictions were put in place to prevent the flow of wealth from England to France. Along with the sanctions came a lucrative black market. In the early 1800’s programmable machines for making lace existed that were almost as sophisticated as modern 3D printers. After World War I, fashion and the position of women in the work force changed and the lace industry faded away. However, the craft is still practiced around the world and pops up from time to time such as at the closing ceremonies for the Rio 2016 Summer Olympics.

When a simple set of actions can be combined to create something complex, it provides an excellent candidate for study by mathematical modelling. The hope is that a small and simple set of rules can be identified that explain how the actions work. For my research, I was able to do just that. I was strongly helped by being skilled in the craft and familiar with how lacemakers describe their patterns. I was able to leverage fairly modern discoveries in mathematics, such as graph theory and braid theory, to describe the rules in a precise manner. Braid theory is just as it sounds — the study of braids such as the kind you make with hair. It first became a subject of interest to mathematicians in the 1860’s when it was believed that different types of atoms resulted from the aether being knotted in different ways — an idea that is not too far from modern String Theory. Graph theory is the study of relationships between pairs of objects. The objects are represented as dots and the relationships are lines between the dots. You can see it used sometimes to describe a network of friends on FaceBook or the structure of links between web sites on the internet. The genesis of this branch of mathematics is a puzzle known as the Seven Bridges of Königsberg.

I started with the idea that little braids, made with four threads, could be visualized as objects and the flow of threads from one braid to the next could be represented as a relationship between the objects.

I then went through a process of converting real world concepts from the craft, such as “lace is a fabric and should hold together as one piece, not fall apart into a bunch of little pieces”, into mathematical rules such as “the graph must be connected” — that is, it must be possible, following the lines of the graph, to get from any dot to any other dot. After identifying each rule, I looked at what kinds of graphs were consistent with the rules set down so far and whether all possible graphs could be turned into lace. If some of the graphs were not valid, I studied them to understand why they would not work and then devised additional rules. I also looked at old lace patterns to see if any of my rules were too restrictive.

Once I was pretty confident that I had chosen the right set of rules, I was able to turn the mathematical description into a computer algorithm. This was not too hard to do because graphs are used quite a lot by computer scientists and I could again leverage the work of others. If I gave the computer any graph, the algorithm was able to tell me whether that graph met all of the rules. I then wrote a computer program that would generate trillions of graph drawings and test them; if the rules were satisfied, the graph drawing was saved as a file. In this way I was able to find over 5 million lace patterns. In the 500 years of lace history, lacemakers have found fewer than 1000 patterns by trial and error. To me this demonstrates the power of combining math and computer science to examine a new problem.

Finally, I selected some of the patterns and turned them into physical pieces of lace by hand. The patterns I chose were ones that I found aesthetically interesting and that had not been made by any lacemaker before me (at least, to the best of my knowledge, the patterns are new). I have only just begun to explore these new patterns — there are very many to choose from. I am also interested in what happens when you break some of the rules. The results won’t look like traditional lace but they could be very interesting.

In addition to applying my research to create art, I am interested in how these new textiles could be used in different ways. We typically think of lace as something ornamental and perhaps frivolous. However, the textiles that can be produced using the techniques of this craft have many interesting properties. Lace is light in weight and full of holes. This could be useful for a material that is used under water in heavy currents; the holes would allow the water to pass through easily but the threads are braided around each other so they hold their shape. The threads travel in many different directions which could be useful for transmitting signals to and from sensors or controllers. The thread directions also give the material unique physical properties when stretched or compressed. Developing new fabrics and materials for high tech needs is a growing area of research. Some of the high performance running shoes by Nike are examples of how companies are applying these new materials today.

*To learn more about Veronika Irvine, try one of these resources:*

Flickr (a photo gallery of Irvine’s work)

http://gallery.bridgesmathart.org/exhibitions/2016-joint-mathematics-meetings/virvine

http://gallery.bridgesmathart.org/exhibitions/2015-bridges-conference/virvine