really knew the earth was round. For example, as a ship sailed away,
you could see its mast receding from view around the curve of the
earth, and Magellan's circumnavigation of the globe would seem to have
settled the question. But is a round earth the only possible
explanation for these phenomena? Actually, if you think about it,
there are many possible forms the Earth could have taken. For
example, suppose our planet were a large torus, or donut-shape.
People would have observed local curvature as they watched departing
ships, and again it would have been possible to travel west for a
while and return from the east. There are a few laws of physics that
tend to produce round planets, but in this podcast we don't care about
such trivial details, we're just thinking about the mathematical
possibilities. Thinking about such alternate possibilities is what
led mathematicians to pose the famous problem known as the Poincare
Conjecture, which was just resolved a few years ago.
Now suppose our planet had been a torus. Would there have been
ways to distinguish that situation from living on a sphere? Actually,
there are various fundamental differences between a torus and a
sphere. For example, if you circumnavigate the sphere, your path
partitions it into two halves, and anyone crossing from north of your
path to the south of it must cross your path at some point. On the
torus, if you were 'circumnavigating' the short way into the
donut-hole and back, your path would not bisect it, and another
traveller could make it to the other side without crossing your path. A
similar test you could do is to stretch a long rubber band along your
path, and tie it together when you return to your starting point:
when you are done, can you slide it along the surface and eventually
contract it to an arbitrarily small size? On a sphere, you will
always be able to, but on a torus, if you traveled into the hole and
back, you will never be able to fully contract the rubber band. The
branch of mathematics that studies basic properties like these of
surfaces is known as topology.
In topology, mathematicians study the essential features of surfaces
that do not vary when they are stretched, or "continuously deformed".
So you can think of a surface as a giant sheet of rubber: tearing or
gluing is out of bounds, but you can distort it all you want. In more
precise terms, two surfaces are homeomorphic, or topologically
equivalent, if there is a continuous, invertible 1-1 mapping between
them. The classic example is that a coffee cup is topologically
equivalent to a donut: both are continuous surfaces with a single
hole. You can imagine creating non-equivalent surfaces by adding
extra 'handles' to a sphere, or punching additional holes in a donut.
Add one handle to a sphere and you have a travel sphere convenient to
take to the airport, but with a little stretching it's also equivalent
to a donut. With two handles you have something equivalent to a kind
of figure-8 donut with two holes, and so on.
A surprising result of 19th-century mathematics was that if you
look at any closed, compact surface that can exist in 3-D space-- that
is, without any infinite protrusions or sharp edges-- it is guaranteed
to be homeomorphic to a sphere with a number of handles, or
equivalently, to a donut with some number of holes. So no matter how
crazy a surface you think you can construct in a 3-D world, in some
sense it is equivalent to a stretched n-holed donut. This result
first appeared in a paper in 1888, though it wasn't rigorously proven
until the 20th century.
Now where does the Poincare conjecture fit into all this? Well,
first we need to extend our vision by a dimension, and think about
discussing three-dimensional surfaces in four-dimensional space. Not
very easy to visualize, due to our daily lives occurring in our lame
3-D universe, but the basic concepts are the same. The question
Poincare asked is essentially whether, just like for 2-D surfaces any
closed, compact surface without a hole is homeomorphic to a sphere, is
any such 3-D surface in 4-space equivalent to a hypersphere? I'm
glossing over a few details here, but that's the basic concept. It
seems like a relatively simple question, but took our best minds over
a century to solve.
You may recall how back in episode 12 I talked about the fact that
this conjecture has now been proven, and about the odd decision of
Grigori Perelman, the eccentric Russian genius who solved the problem,
to refuse the Fields Medal. At the time, I wimped out of trying to
describe the theorem itself-- but since then I read an excellent book
on the topic, by Donal O'Shea, which gave me enough basics to attempt
this podcast. If you're still confused, an entirely likely
possibility, I highly recommend taking a look at that book, which is
linked in the show notes.
And this has been your math mutation for today.