by Julia Collins
Yesterday I was lucky enough to meet with the queen of mathematical crochet: Professor Hinke Osinga. Hinke became famous in 2004 when she crocheted an amazing mathematical structure called the Lorenz manifold - a complicated surface visualising a chaotic system of differential equations. It contains over 25,000 stitches and took four months of work to complete, and though Hinke must have described its details to hundreds of people before me she seems as excited about it as when it was first completed.
"This wire running up the middle is the z-axis", she explains. "And this other wire across the middle indicates a very special set of points. Particles on this line approach the centre by a different method to anywhere else on the surface."
The Lorenz equations themselves describe the path of particles in a system, such as an air particle moving in a weather system or a molecule moving in a chemical reaction. They are called a chaotic system of equations because very small differences in the position of particles at the beginning can result in vastly different outcomes at the end - hence the idea of the 'butterfly effect', where a butterly flapping its wings in New Zealand could set off a hurricane in Texas. Most particles following the equations will end up following a particular trajectory, called the Lorenz attractor. More interesting to Hinke were the special particles which ended up getting sucked into the origin, or centre, of the system.
In her paper Hinke has a great way to describe what her crocheted surface is all about. "Imagine a leaf (a point in the system) floating in a turbulent river and consider how it passes either to the left or the right around a rock (the origin) somewhere downstream," she says. "Those special leaves that end up clinging to the rock must have followed a very unique path in the water. Each stitch in the crochet pattern represent a single point that ends up at the rock."
I asked Hinke whether the act of crocheting the manifold helped her to understand the mathematics of it better. "I think I learnt the most from writing the algorithm to create the crochet instructions", she says. This code, written together with her partner Professor Bernd Krauskopf, computed the stitch pattern incrementally, travelling outward in circles from the centre. "But it helped us to understand the curvature of the surface, and we were inspired in our project by the work of Daina Taimina and her invention of hyperbolic crochet". In Daina's hyperbolic planes the curvature is the same everywhere, meaning that the crochet instructions are very simple: just add in new stitches at a constant rate, for example, every fourth stitch. But in Hinke's surface the extra stitches come in a more complicated pattern, resulting in a more interesting structure but also requiring a lot more concentration to make!
Hinke's website has a wonderful section containing pictures of crocheted Lorenz manifolds from around the world. She originally offered a bottle of champagne for the first manifold to be made, and was delighted by the number of people who took on the challenge. One of the first photos is from Craig Lazarski, who uses the manifold as a playground for his crocheted chickens. Another favourite story I heard was of a man who got halfway through the project but then had his crochet stolen when he left it sitting in a pub one day while he used the toilet. We have to wonder what the intrepid thief did when they realised what they had taken!
To hear more stories from Hinke and to see her amazing Lorenz manifold in person, come along to her talk on Saturday 3rd September at 4pm. If you can't wait until then, you can read more about the project, and get the crochet instructions, from Hinke and Bernd's paper.