Why is the
hyperbolic plane important historically?
DH: The discovery of the hyperbolic plane came from the attempt
to prove Euclid’s fifth postulate, which is also known as
the parallel postulate. For two thousand years mathematicians accepted
this as true, but it had always caused problems.
DT: Mathematics is supposed to be an axiomatic system, but the
fifth postulate does not look like an axiom, rather like a theorem.
And so there were many attempts to try to derive the fifth postulate
from the others. Nobody was able to do that. So the question eventually
arose, well what happens if this postulate is not true? When mathematicians
like Gauss and Bolyai and Lobatchevsky started to take that idea
seriously, they discovered there was a geometry in which all of
Euclid’s postulates hold true except for the fifth one.
Spherical geometry is pretty easy to understand
because we see spheres all around us. But when mathematicians first
started to study hyperbolic geometry they didn’t have any
idea what this space might look like and they were nearly driven
mad trying to understand this space. In 1997, Daina, you worked
out how to make a physical model of the hyperbolic plane using crochet.
How did that discovery come about?
DT: For the past 125 years or so mathematicians had conceptual
models of the hyperbolic plane, such as the Poincare disc model,
developed by the French mathematician Henri Poincare. Some of the
models had great aesthetic appeal, especially through the enormous
variety of repeating patterns that are possible in the hyperbolic
plane. For instance, after the geometer Donald Coxeter explained
these conceptual models to Escher, he used patterns based on these
models in several of his prints.
But these were all conceptual models. Many students and mathematicians,
including us, wanted a more direct experience of hyperbolic geometry
- an experience similar to handling a physical sphere.
In 1868 the Italian mathematician Beltrami had described a surface
called a pseudosphere, which is the hyperbolic equivalent of a cone.
He actually made a version of his model by taping together long
skinny triangles - the same principle behind the flared gored skirts
some folk dancers wear. The in the 1970s the American geometer William
Thurston had described another model of hyperbolic space that could
be made by taping together a series of paper annuli, or thin circular
strips. All these models were time consuming to make and hard to
handle; they are fragile and they tear easily. I realized that Thurston’s
construction could be made with knitting or crochet - basically
all you’d have to do is increase the number of stitches in
each row. I grew up in Latvia doing these handicrafts and I decided
to try and make one. At first I tried knitting, but after a while
you had so many stitches on the needles it became impossible to
handle. I realized that crochet was the best method.
I have crocheted a number of these models
and what I find so interesting is that when you make them you get
a very concrete sense of the space expanding exponentially. The
first rows take no time but the later rows can take literally hours,
they have so many stitches. You get a visceral sense of what “hyperbolic”
really means.
DT: We use these models a lot in our classes at Cornell. Most
of our students don’t make the models themselves but they
get a physical sense of the properties of the hyperbolic plane from
studying them. One thing is that you can physically experience straight
lines. You can fold the crochet models and see how straight lines
behave and how they intersect. It really helps the students to understand
very quickly the intrinsic properties of hyperbolic geometry.
DH: Another thing is that you can see the properties of triangles,
particularly what’s called the ideal triangle. On a Euclidean
plane the internal angles of a triangle sum to 180€, but on a hyperbolic
plane they always sum to less than 180€. Moreover, as the vertices
of the triangle get further away, the interior angles approach zero.
When the points are at infinity, which is the largest triangle you
can draw on the hyperbolic plane, the interior angles sum to zero.
This is one of the exercises we give the students, and it’s
very mysterious because the area of any ideal triangle on a hyperbolic
plane will always be πr2, where r is the radius of curvature
of the plane. So on any particular hyperbolic plane all ideal triangles
are congruent - they all have the same area - which is pretty amazing
I think.
What do other mathematicians think of this
incursion of feminine handicraft into their domain?
DT: When they see the pictures in our book lots of people want
to have models themselves. I’ve made them for mathematics
departments all over the world. There is one in the Smithsonian
in the American Mathematical Model Collection. But they are time-consuming
to make and it’s tough on your hands because you have to hold
the wool very tightly to get a good stiff tension with the crochet.
These days people have to want one very badly and keep on calling
me, then maybe I will make them one. In January 2005 at the Joint
Mathematics Meetings in Atlanta there will be a special session
on Mathematics and Mathematics Education in Fiber Arts in which
I will be giving a talk on these models. |