by Joyce Gramza
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."
Taimina Knit Researcher
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.
Knits on Table
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
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.