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January 26, 2013
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Extreme Knitting


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  Hyperbolic Geometry

Learn to Knit and Crochet



   06.22.06
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Along with sunscreen and a book, knitting needles and yarn have become a staple of many vacationers' beach bags. Now, as this ScienCentral News video reports, it turns out that knitters might hold in their hands the secrets of the brain, or the universe itself.

A Strange Yarn

Most people who knit or crochet know what their finished product will look like. But you never know what's going to come from Daina Taimina's crochet hook. And that's the point, says her husband, David Henderson, as he holds up a convoluted piece that resembles lettuce, or a human brain -- if lettuce or brains were purple, that is.

"It actually looks like a brain," Henderson marvels. "And I had no idea it was going to look like that until Daina did it."

The couple, both Cornell University mathematicians, study and teach about the strange world of hyperbolic space, in which everything constantly curves away from itself. In hyperbolic space the most basic shapes in regular geometry, like a plain old flat plane, are warped into hard to imagine shapes.





"If you have something flat, like a flat floor, flat top or flat tabletop, so then that's a zero curvature, nothing is curved," explains Taimina. "If there is a positive curvature, constant positive curvature that's when we get a sphere. And now the question is, if there is a negative constant curvature, what is it? That's a hyperbolic plane. So in some ways we can say it's opposite of a sphere."

If that sounds mind-boggling, it's exactly why mathematicians need models, the couple explains. "There is no equation for a hyperbolic plane," says Taimina. Because, well, that's the point of these models. You can have some object which doesn't have an equation."





"I had been studying hyperbolic geometry for a long time, but until this actually got crocheted, nobody could describe what it was going to look like," Henderson says. "There it is -- I still can't describe it, I just have to say, 'Look!'"

Taimina, whose hyperbolic crocheted objects are displayed in art exhibits around the world, learned to knit as a girl growing up in then-Soviet Latvia, where it wasn't a craze, but a tradition. "In school, in one way it was to do things like trying to do several things at a time, kind of like read a book and knit, or watch TV and knit," she says. "And also the way to get some things which nobody else has, because once you knit your sweaters, that's unique."

She had the unique idea to crochet a hyperbolic plane when Henderson persistently played and taught with a paper model long after it was in tatters. "So I said well, if it can be made out of a paper ... I can crochet it. And then it's easier to handle and then you can much more successfully play with it and it won't fall apart, and so it's more useful. So that's how it all started," Taimina says.

She creates her strange and beautiful pieces by constantly increasing her number of stitches in each row by a certain amount.




Now Henderson can play with them all day long -- folding them to find straight lines, parallel lines and intersections, twisting and turning them inside out to discover the relationships between what otherwise would seem like wildly different shapes, or seeing why a structure that looks a lot like lettuce is also a great way for nature to store information in a human brain.

Exploring the convoluted folds of the ruffly purple brain-resembling piece, "it's hard to tell how far apart one point is from another, or which direction to go to get somewhere," he points out. Then Taimina finishes his sentence for him (something they do a lot of): "But all points are no more than 10 inches apart. So to get from one place to another it's very quick, you move back into the center and then back out again."

"So I can experience what the geometry actually is in the hyperbolic plane," Henderson says. "Otherwise it was just abstract things with abstract representations for it, but now it's something we can actually feel and see and experience directly. Like a hyperbolic geometry lesson. That wasn't possible until these models."

And now that people know what they look like, hyperbolic shapes can be recognized -- and studied -- all over the place in nature. "We are not used to seeing them in nature, we don't know how they look and that's why we don't know what to look for," Tiamina says. "Once you get more familiar with them -- and that's possible with these models -- you know what to look for and then you can just start to see, 'oh there are things I have seen!' Certain flowers opening that way ... certain kinds of mushrooms ... some sea slugs ..."

In fact, since many of the shapes turn out to be used by marine life, Taimina's source of funding, The Institute for Figuring, has a volunteer project in which anyone is invited to contribute to crocheting a model coral reef with all its diverse organisms.

For not only are crocheted hyperbolics now being put to use in math classes everywhere, some non-mathematical whizzes figured out they were art. "People who were non-mathematicians, they said, 'Oh, but that's an art,'" Taimina laughs. "And then we just said like, 'No, this is not art, its mathematics,' and they said, 'No, these forms are art'!"

Oh, and by the way, the couple points out that hyperbolic geometry is also interesting because it just might hold the secret to the shape of the universe.
Other practical applications besides topology, biology, information storage and computing include understanding how the Internet behaves and improving digital animations for movies and video games. For example, says Henderson, "It's very difficult to get clothing so it looks naturally draped, and that's because you can't write equations for it. So this is a way of studying how to do it."

Taimina's practical application -- a hyperbolic skirt that took her three months to crochet using more than a mile and a half of yarn -- may not take the fashion industry by storm. Then again, stranger yarns than that have happened, and it sure is twirly.

David Henderson and Daina Taimina wrote the book, "Experiencing Geometry: Euclidean and non-Euclidean with History," Prentice Hall, 2005, and their work was featured in "Discover" magazine, March 2006. Their research is funded by The Institute for Figuring.


 
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