Crochet Mathematics: Chaos

(c) Wikipedia

This is probably a pretty cool design you've seen on a t-shirt, perhaps while perusing thinkgeek.com (one of my favorite geek-gear websites). But what is it? Whelp, I'm here to tell you AND show you how it can be represented through crochet.

In 1963, Edward Lorenz, a meteorologist, introduced the Lorenz attractor which was a form derived from the simplified equations of convection rolls arising in the equations of the atmosphere.

And you say... huh?

Me too - and this is where the research begins.

The Concept
The idea is that chaos is derived from a series of questions that help describe the chaos of nature. These equations, the Lorenz equations, have been used to help describe the behavior of chaotic systems like weather and water turbulence. Lorenz first developed a simplified model of thermal convection. He used three different equations to model two thermal convection rolls, each behaving unpredictably. He noted that even the small change in the beginning caused the system to evolve very differently in a short amount of time; however, while unpredictable, the equations produced consistent structure.

So, what in the world does all this mean?

If you put a gaseous substance into a box and heat it up, you will note that the warm gas will rise and the cooler parts will sink. As the gas closest to the walls warms it will heat and rise, and as it rises it will cool and fall again. This will form cylindrical rolls. With a regularly applied temperature, the moving gas should be regular and predictable; however, its not even if it appears to be. The movement is chaotic. The gas, even under the conditions described, will roll for a while in one direction and then stop and reverse direction. It will continue to randomly reverse directions at unpredictable time and speed.

Why? Because regardless how smooth a surface may appear, at a molecular level it is not smooth. Something at a molecular level can impact the overall pattern.

Think to the butterfly effect - with Lorenz's observations, the beat of one butterfly wing can change weather patterns drastically - perhaps not immediately, but it can cause change that otherwise would not have occurred.

(If you are interested in seeing the equations in full, you can click here to view.)

Crocheting the Concept
Dr. Hinke Osinga and Professor Bernd Krauskopf, both of Bristol University, got the idea to crochet chaos using computer generated patterns sketching out the increases and patterns created by the Lorenz equations. Osinga, who had learned to crochet at age seven, took the computer generated pattern and got to work. The pattern could always be observed in a 2D image, such as one on computer screen, but being able to view it in a 3D world, via crochet, was exciting for the researchers.

Lorenz Manifold

After crocheting just over 25,000 stitches in 85 hours, Osinga has created chaos. The crocheted model, called the Lorenz manifold, helps visualize how chaos arises and is organized in systems like chemical reactions and biological networks.

Dr. Osinga explains this shape, this concept by imagining "a leave floating in a turbulent river and consider how it passes either to the left or to the right around a rock somewhere downstream. 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 represents a single point [a leaf] that ends up at the rock."

Osinga and Krauskopf insist their crocheted manifold was not done for fun (although any crocheter and mathematician/math geek may disagree), they insist their work gives true insight on how chaos arises and that chaos is organized in systems. While I still insist it had to be fun, I cannot imagine how wonderful Osinga had to feel as a concept viewed only on a computer screen came to life as she created each stitch.

Oh, want to know something even more exciting for those truly intrigued by the shape of chaos? You can crochet your own following the original instructions published by Osinga and Krauskopf (click here if you want to give it a whirl).

Just remember, if you tackle crocheting chaos, I need to see some pictures!

References:
The Guardian http://www.guardian.co.uk/education/2004/dec/16/research.highereducation3

MathsReach http://www.mathsreach.org/Crocheted_Chaos

Math World http://mathworld.wolfram.com/LorenzAttractor.html 

Physorg.com http://www.physorg.com/news2395.html

Wikipedia http://en.wikipedia.org/wiki/Lorenz_attractor

Zeuscat http://www.zeuscat.com/andrew/chaos/lorenz.html