Crocheted Chaos

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The piece (right) represents part of a surface in the classic Lorenz chaos system, constructed in 1963 by USA meteorologist Edward Lorenz in
an attempt to understand why the weather was so hard to predict. Lorenz developed a simplified model of thermal convection in the atmosphere - as hot air rises, cools and falls it forms convection rolls, which can sometimes be seen as long lines of parallel cylindrical clouds. Lorenz used three differential equations to model two neighbouring convection rolls. He was surprised to find that the equations behaved unpredictably:
the smallest changes in starting co-ordinates led to very different evolution of the system after only a short time.

Despite their unpredictability, the equations produce a consistent structure. Whatever the starting co-ordinates, the system is attracted to spiralling sets of curves that form a butterfly shape with two wings. The motion on this chaotic Lorenz attractor is very unpredictable and continually swaps from one wing to another.

With her collaborator, Professor Bernd Krauskopf, Osinga tried to look at the system differently, to obtain “a more static, time independent image of what is going on”. Using computer visualisation tools, they identified and animated a special surface of the system associated with the origin, where x = y = z = 0. Called the stable manifold of the origin, points on this surface do not go to the chaotic attractor, but converge to the origin instead. The system’s chaotic dynamics are expressed through the complicated geometry of this special surface. “It is incredibly difficult to decide which points lie on it - and so don’t exhibit chaos - and whichpoints do not,” says Osinga.

Krauskopf and Osinga spent years developing a computer algorithm that accurately constructed these surfaces. Then, suddenly, the pair realised that “the way that we computed the surface naturally translated into crochet instructions. When I saw that, I just had to try.” Osinga spent 85 hours crocheting more than 25,000 stitches, producing a metre-wide piece of the pancake. The form is held in shape by three wires, which represent the vertical z-axis; all the points with the same distance from the origin along the outer rim; and the only two solutions that meet perpendicular to the z-axis at the origin. “One major advantage of making a real crocheted model of the manifold is that it gives a better idea of the size of chaotic systems in real life than the tiny animations on the screen,” she says.

“At the top you have a helical rotation going up, and horizontally there are two spiraling rotations going in opposite directions. Chaos is everywhere, which means that this ‘pancake’ folds over and fills space all around us. It is very, very long - you can think of it as a space-filling pancake, much like the examples we know of plane-filling curves.” The release of the crochet instructions in the Mathematical Intelligencer in 2004, which is read in many universities, colleges and high schools around the world, stimulated a global rash of crochet and sculpture about chaos. The crocheted Lorenz manifold has become an art object as well as a useful teaching tool.

See also
Mathematical Intelligencer crochet instructions:

Above: Osinga and Krauskopf provided these pictures of the manifold for London Underground posters in World Mathematical Year 2000.