... Predicting the weather remains an inexact science. Despite recent advances in technology and considerable investment in research, the patterns of clouds and the effects they have remain to an extent a mystery. Modern science may have sent people to the moon and mapped the hidden world of subatomic particles, but the weather has stubbornly refused to reveal its secrets. One of the first scientists to investigate this puzzling mutation of science was an American meteorologist called Edward Lorenz. His research would lead to the development of the mathematical field of chaos theory. Chaos theory is about pattern, how patterns occur in nature, and how surprisingly complex this process can be. One example he looked at was that of atmospheric thermal convection, which he attempted to model using differential equations. You get this motion of the hot air rising and then it kind of spreads out and cools back down. You get what are called convection rolls. And you can actually see those sometimes when the clouds really form this cylindrical shapes, these long lines and several clouds next to each other. These are these convection rolls that Ed Lorenz was trying to model. Lorenz developed the equations named after him to model two adjacent convection rolls. Their initial condition can be described using three coordinates, X, Y and Z. This model is called the Lorenz attractor. What Lorenz discovered when he studied these models was that this point gets pushed around and it actually never comes back to itself. As you would be going around in circles several times and then suddenly you changed direction and you were counterclockwise. And it was impossible to predict when you were going to switch, which role you're in, how many times you're going to go around and then switch over to another one. The smallest change in the starting conditions, even though smaller than can be measured, led to very different evolutions of the system after a short period of time. This is the definition of a chaotic system. Ben Krauskoff and I, we got interested in these Lorenz equations thinking like, well, there is another way of looking at the system rather than just simulating, just seeing how things run over time. We try to get an image, a more more static time independent image of what is going on. And we came up with the notion of the Lorenz manifold, as we call it. There's a surface of all points in space that don't go to this convection role motion. They are actually defined by the fact that they are going to one special point in space, the origin. So whenever you pick an initial condition in space and it's on the Lorenz manifold, you end up at the origin. The origin is where x, y and z are all zero. It took quite a long time before we realized you could just read off what the computer is doing and follow it in crochet. The final result took more than 25,000 crochet stitches and about 85 hours to complete. And then we ended up with this huge pancake. It looked nothing like the Lorenz manifold because in the real world we have gravity. So when we were thinking of how to mount the surface, we were kind of being a bit picky. As mathematicians, we wanted each wire to have special mathematical meaning as well as holding up the surface in the correct shape. And so there are three wires, special wires going through this object. There's the outer rim and that's just because of how we calculate the surface on the computer. The outer rim consists of all the points that have the same distance to the origin, which is the center of this object. Now these also have very special meaning because they are the two solutions. So when you follow them, they go to the origin. So one side here goes to the origin and the other side coming from the other end goes to the origin as well. And they, as you see, come in perpendicular to the z-axis. So if you would pick any other point on this surface and you would look at the solution, how it gets pushed towards the origin, everything gets pushed towards the z-axis and then comes in tangent to the z-axis. So these two solutions are the only two that don't do that. The crochet pattern Hink and Bern published in the Mathematical Intelligence in 2004 has inspired many people to crochet their own manifold and has provided a new and novel way of illustrating a chaotic system.