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“For God’s sake, please give it up. Fear it no less than the sensual
passion, because it, too, may take up all your time and deprive you of your
health, peace of mind and happiness in life.” Wolfgang Bolyai

AH, THE perils of hyperbolic geometry. Bolyai, a Hungarian mathematician, was
so frustrated by this devilishly difficult branch of mathematics that he begged
his son Janos not to take it on. If only he’d lived 150 years later. Daina
Taimina, a visiting professor at Cornell University, has tamed the wild beast of
hyperbolic geometry using nothing more mathematical than a crochet needle and
some yarn.

She’s just one of a growing number of mathematicians who use the power of
traditional handicrafts to create otherwise unimaginable objects. If you have
the right skills, you can knit an unending surface, or stitch together a
hyperbolic soccer ball. You can even blow a glass bottle that has no inside and
only truly exists in four spatial dimensions. When your parents told you that a
hobby can add a new dimension to your life, they really had no idea.

Although, like Bolyai, you might not want your children to explore the
properties of hyperbolic geometry, someone’s got to do it. It could be vital in
working out how the world works, and how we perceive it.

Hyperbolic geometry is one of the three possible geometries for describing
the shape of the physical Universe (the other two are Euclidean and spherical
geometry). It describes things that exist in curved space, rather than the
flatness of the more familiar Euclidean geometry. Hyperbolic geometry is also
important in visual perception. Wayne Zage, a computer scientist at Ball State
University in Muncie, Indiana, has shown that our binocular vision actually
perceives things in hyperbolic space. And it could prove vital to brain
surgeons: Monica Hurdal of Florida State University is creating hyperbolic maps
of the brain that enable surgeons to understand which areas are which before
they begin to cut bits out. Startlingly, examples of hyperbolic planes even
exist in nature, on—among other things—the surfaces of some lettuces
and seaweeds.

It may be important, but hyperbolic geometry is mind-bendingly hard to
fathom. If this piece of paper were a hyperbolic plane, rather than a
straightforward Euclidean one, it would be curved both up and down at every
point (like the mid-line of a saddle). As well as making it hard to read, that
would give it some strange properties. Perhaps most bizarrely, you could draw
infinitely many lines that run parallel to the right-hand edge of the page, all
of them passing through this full stop. On the Euclidean pages of New
Scientist, that’s impossible.

Is that clear? Probably not. Which is exactly Taimina’s point. Words aren’t
enough: you need real, physical models of this kind of thing to play with. You
can’t really understand the hyperbolic plane until you’ve held one in your
hands.

Taimina encountered this geometrical beast in her native Latvia. “When I
first learned about hyperbolic geometry I think I more believed than I could
imagine,” Taimina says. But now her faith has been replaced by a crochet
hook.

Her venture into geometrical craftwork was inspired by David Henderson,
another professor at Cornell. He made his first hyperbolic plane in 1978, using
a Swiss Army knife and a lot of patience to cut one out of paper. Four years
ago, Taimina attended one of Henderson’s workshops and saw him using another
paper plane. It seemed too fragile to really get to grips with, so she began to
think of ways to make a more durable version.

The first thing she tried was knitting, but the resulting plane was too
floppy. Undaunted, she tried crocheting—and it worked beautifully.

The simple act of crocheting a hyperbolic plane has led to new discoveries in
this field of geometry. “David found a formula for the areas of ideal triangles
on hyperbolic planes, just because he looked at this model,” Taimina says. It’s
been such a success that earlier this year, Henderson and Taimina shared their
techniques in The Mathematical Intelligencer(vol 23, p 17). Now you too
can crochet yourself a hyperbolic plane. All you need is a few basic
crocheting skills—but remember to keep the stitches tight and even.

The idea of “mathcrafts” is catching on. People have started turning up to
Henderson and Taimina’s Cornell workshops with their own crochet hooks. And at
the joint American Mathematical Society/American Mathematical Association
meeting in New Orleans last year people were even getting together for a
knitters’ circle.

There’s more to it than hyperbolic planes. You can knit, crochet or sew all
sorts of mathematical things. And you can even make money from it. Cliff Stoll,
for instance, a lapsed astrophysicist living in Oakland, California, pays
several smiling grandmothers, a former librarian, a nurse and a schoolteacher to
knit strange topological hats.

The hats are derived from the Möbius loop. Take a strip of paper, give
it a half-twist and tape the ends together. Despite covering every inch of the
paper and no matter how far it goes, an ant walking on this loop will only ever
find one surface.

In 1882, the German mathematician Felix Klein imagined joining two
Möbius loops together: this, he realised, would create a bottle whose
inside was also its outside. But he had to be content with imagining it because
a true Klein bottle requires four spatial dimensions—it has to pass
through itself without making a hole.

Nonetheless, it is possible to produce an approximate Klein bottle in our
physical Universe, and that’s what Stoll—and his knitting
circle—have learned to do. He made his first Klein bottle in glass. “It
took skill, patience and burnt fingers,” he says. But eventually, he managed to
stretch the neck of a bottle through its side and join its end to a hole in the
base.

A mathematician friend of Stoll’s thought it was the best thing he’d ever
seen. “So he bought it from me, and I made another one,” Stoll says. Another
mathematician bought that one. “So I thought I’d make a whole bunch of them.”
And so a business was born. Last year he also launched a range of Klein bottle
beer glasses, called Klein Steins, that can hold a foaming pint despite having
no inside.

Hats are his latest invention, developed in conjunction with knitwear artists
Eleanor Kent and Betty Lampen
(see diagram, right).
Like ordinary woollen hats, they keep
your head warm, but they have a few other advantages, Stoll says. For a start,
like the bottles, they have no inside or outside. “You might think that would be
useless, but it has a lot of advantages,” Stoll says. “With only one surface,
they’re easy to keep clean.”

He’s joking, of course. Unlike Taimina and Henderson, Stoll isn’t trying to
extend the boundaries of mathematics or our understanding of the Universe. He
doesn’t even make a living from it. It’s just fun, he says. He can afford to
play around because, in the 1980s, he caught KGB spies hacking into US military
computers. Once they were convicted, Stoll wrote his story in a couple of
bestselling books that earned him all the money he’ll ever need.FIG-mg23225401.jpg

So now he just fiddles around with mathematical handicrafts. “When I tell my
neighbours what I do, I just get this strange look,” Stoll says. His topological
creations act as a kind of mathematical divining rod. “You either get a ‘wow!’
or a ‘huh?’,” Stoll says. “That’s the dividing line between people who have
studied math or topology and those who haven’t.”

But maybe Taimina, Henderson and Stoll could inspire future generations of
mathematicians. If you’re concerned about your children’s grades at school,
don’t lecture them—grab your knitting needles and a ball of wool. Those
Garfield slipper socks can surely wait.

* * *

To crochet the hyperbolic plane…

1. Make your beginning chain stitches
   (see diagram, left). About 20 chain stitches
   for the beginning will be enough. Join this row into a circle.
2. For the first stitch in each subsequent row, insert the hook into the
   second chain from the hook. Take yarn over and pull through chain, leaving two
   loops on hook. Take yarn over and pull through both loops. This completes one
   single crochet stitch
   (see diagram, centre).
3. Before doing your second row, you must choose the radius of your hyperbolic plane.
   Pick a number, N. If N is large, the radius of curvature will be large,
   and your plane will be fairly flat. If it’s small, you’ll get a much more curved plane.
4. For the next N stitches, insert the hook into the next chain instead
   of the second.
5. For stitch N+1, proceed as before, but insert the hook into the same loop as the
   Nth stitch.
6. Repeat Steps 3 and 4 until you reach the end of the row.
7. At the end of the row, before going to the next row do one extra chain stitch.
8. When you have the model as big as you want, stop by pulling the yarn through the
   last loop.

With thanks to Maureen Frier for testing the pattern