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Crocheted Manifold


A close-up view of the crocheted Lorenz manifold. The origin, the center of the bulls-eye pattern on the right, is just hidden from sight. The wire looping through the origin is the strong stable manifold of the system. The manifold's vertical axis of symmetry can be seen as a diagonal across the upper half of this image. Photo: University of Bristol, used with permission.


Osinga and Krauskopf with their model of the Lorenz manifold. Photo: University of Bristol, used with permission.

As Daniel Engber reported in the Chronicle of Higher Education for January 21, 2005, a team at the University of Bristol has used yarn and a crochet hook to build a model of the Lorenz manifold. This is the 2-dimensional stable manifold of the origin in the Lorenz system

x' = σ(y - x)
y' = ρx - y - xz
z' = xy - βz

with the classic choice of parameters σ = 10, ρ = 28, and β = 8/3. According to Engber, Hinke Osinga and Bernd Krauskopf realized that the computer program they had devised for generating the Lorenz manifold could be adapted to produce a set of crocheting instructions. "Each computed point on the manifold translates to a type of crochet stitch. A mere 85 hours and 25,511 stitches later, the project was finished." Osinga and Krauskopf's work appeared in the fall issue of The Mathematical Intelligencer ; their preprint is available as a PDF file online. The crocheted Lorenz manifold struck the fancy of the international media, including the BBC (Mathematicians crochet chaos), CBC Radio (Crocheting Chaos), the Austrian ORF, and Channel One in Russia.

 

"Blood, math and gore."

It could work." That's the end of Alessandra Stanley's review of the new TV series "Numb3rs," in the January 21 2005 New York Times. The plot line involves "Don, a decent, workaholic F.B.I. agent who turns to his math genius younger brother, Charlie" for help in tracking down a serial rapist. As Stanley tells it, "Charlie looks at a water sprinkler and has an Archimedean moment: he realizes that the same principle that allows him to track the path of drops to determine their point of origin could be applied to the distribution of crime scenes on a map." (She quotes one character as saying: "If this works, we'll have a whole new system of investigating criminal cases.")

More about "Numb3rs"

A more academic view was taken by NPR's "Math Guy" Keith Devlin, interviewed by Scott Simon on "Weekend Edition - Saturday" for January 22, 2004. Scott: "There's a scene where the mathematician brother is writing out a formula on the board. Firstly he seems to be listening to head-banging rock music and in addition to that he seems to be in the grip of a fever. Is that commonly what happens when mathematicians write out formulas?" Keith: "... Most people's impression of a mathematician, if that impression is of an elderly guy in a tweed suit and worn down shoes, they'd better walk around a university like Stanford or Cal Tech or MIT and just take a look. In fact when David Krumholtz was preparing for this role, he hung around Cal Tech for a while and just watched what he saw."
   Scott plays a clip in which Charlie consults a fellow mathematician who tells him: "Charlie, when you're working on human problems, there's going to be pain and disappointment." Keith: "This reflects one of the most interesting changes in the whole history of mathematics. ... Over the last few hundred years increasingly we've found that we can take this mathematics which was originally developed to study the physical world and apply it to the world of people, and by using computer graphics superimposed on action you can show people that mathematics, this abstract stuff, really applies to the real world and, in the case of a crime series, with positive outcomes for society."
   Scott: "Do you expect that this series could do for mathematics what 'The Simpsons' did for cartoons?" Keith: "I would hope it does succeed because the one thing they're trying to do is make mathematics look cool. I know it's cool, all my friends know it's cool. We do have an image problem, and I think a TV series like this can help get over it." The interview is available online.

Terror Network Theory

On December 11, 2004, Jonathan Farley was interviewed on Air America's "So What Else Is News" by the program host, resident whiz-kid Marty Kaplan. Farley, currently a Visiting Scholar at Harvard, turns out to be a mathematician with a mission.

 

Removal of four nodes at random has a 93% chance of disconnecting a 15-node binary tree, but only a 33% chance of breaking all top-to-bottom chains of command.

Inspired by the real and hypothetical mathematical derring-do evoked in "A Beautiful Mind," he has found an application of lattice theory to the war on terror. His problem is the structure of terror cells and what it takes to disrupt them. A current approach, he tells us, is to view a terrorist cell as a graph, "a picture where you've got a bunch of nodes or dots which represent the individuals, and then lines which connect individuals if they have some sort of communications link, or if they lived in the same flat in Hamburg at some time ..." Graph-theoretically, a cell is disrupted if the graph is disconnected. Farley noticed that with that kind of analysis "you're missing a key mathematical component of the terrorist network, namely its hierarchy. And that's where I come in, because my branch of mathematics, called lattice theory, deals with hierarchy and properties of order." Kaplan proposes a concrete example: suppose a cell has 15 people, "and the government has picked off 4 of them. To what degree can the government feel as though they have shut that cell down?" Farley explains that for a precise estimate you would need to know the structure of the cell, but he shows how, for a 15-node binary tree, hierarchically ranked from top to bottom, the graph-calculation and the lattice-calculation give very different answers. "If you've captured 4 guys you're pretty sure you've disrupted the cell, under the old way of thinking. But when you take the lattice-theoretic perspective, you see that actually you only have a 33% chance of disrupting the cell in that case." He elaborates: "If 4 people have been captured at random, it might still be possible for terrorist plans to be passed on from the leader down to one of the people at the bottom, one of the eight foot soldiers, in which case you might have another September 11, you might have a shoe-bombing ..." And finally: "Mathematics won't help you catch the terrorists, but it will help you analyze how good a job you've done in the past." Farley's work has also been covered by Ivars Peterson in Science News Online (January 10, 2004).


Tony Phillips
Stony Brook University
tony at math.sunysb.edu