ÐÓ°ÉÔ­´´

A series of pictures can be as convincing as a series of equations.
These pictures are part of a video of a proof of a special case of the Thurston
conjecture, one of the outstanding problems of topology named after William
Thurston at Princeton University.

To a topologist, a coffee cup is the same as a doughnut. Each of them
has a hole (the coffee cup’s hole is its handle), and, if they were made
of putty, you could remould one into the other without cutting or tearing
it. In topology, the concept of a straight line or the measurement of an
angle or area do not make sense because they are changed by deformation.
By contrast, straight lines and angles are what geometry is all about. Topology
is floppy, geometry is rigid.

More than a century ago, Bernhard Riemann realised that any floppy topological
surface could be given a geometry. An easy example is the surface of a balloon.
To the topologist, a partially inflated, floppy balloon is the same as the
surface of a rigid, perfectly round ball – all you have to do is to puff
up the balloon gently to change one surface to the other. Geometrically,
the two are quite different but by inflating the balloon it acquires a characteristic
geometry. (In this spherical geometry, for example, the shortest distance
between two points is a great circle. And every triangle has more than 180
degrees.) By finding a geometrical structure on a topological space, mathematicians
make the space rigid and provide themselves with measuring tools that can
be used to prove difficult results.

Thurston suggested that it may also be possible to impose geometries
in three dimensions. He conjectured that each three-dimensional space can
be cut up into pieces in a standard way, so that each piece can be ‘inflated’
to have a geometrical structure. Thurston has proved his conjecture for
many different types of space, in particular for the space obtained by removing
a particular group of linked circles from the ordinary three-dimensional
space in which we live. If his conjecture were proved in all cases, then
the notorious Poincare conjecture, one of maths’ outstanding problems, would
follow as a special case.

Unlike a balloon’s latent spherical geometry, most three-dimensional
topological objects acquire three-dimensional hyperbolic geometry. This
is a geometry in which the sum of three angles of a triangle is always less
than 180 degrees and in which there is so much room that, if the standard
unit of length is 1 metre, then a hyperbolic hemispherical swimming pool
50 metres across would contain 23 times the volume of our planet. The hyperbolic
plane is famous for providing a disproof of the 2000-year-old Euclidean
parallel postulate: given a line L and a point P not in L in the hyperbolic
plane, there are an infinite number of lines through P which do not meet
L (are parallel to L).

The pictures above are from a video called ‘Not knot’, created on a
computer at the new Geometry Center in Minneapolis by Charlie Gunn. The
video is a ‘proof by pictures’ of a special case of the Thurston conjecture;
it shows that the complement of the Borromean rings (what is left when the
rings are taken away) has three-dimensional hyperbolic geometry. The Borromean
rings are the heraldic symbol of the Princes of Borromeo, and they are carved
in the stone of their castle on an island in the beautiful Lake Maggiore
in northern Italy. The three rings have the property that they are linked,
but that if any one of them were removed, the other two would fall apart.
They are sometimes said to symbolise the Holy Trinity and sometimes the
motto ‘Unity is strength’.

The picture at the front of the article shows what it would look like
to live in the space where the three Borromean rings were coloured red,
blue and green, and the rings are axes of order two symmetry. Two-fold symmetry
means that as you walked the short way round a bracelet, what you saw would
be repeated half-way round. The resulting geometry is Euclidean. If this
grid of lines looks nothing like your idea of a triplet of rings, remember
that you are not looking at the rings from a position outside. You are looking
at a universe composed entirely of the complement of the rings themselves,
from the inside.

In all the other pictures the geometry is hyperbolic. They were created
by Gunn using lighting and physics appropriate to a hyperbolic universe.
Thus light intensity falls off, not by the Newtonian inverse square law,
but much faster, by the reciprocal of the square of the hyperbolic sine
function. The second and third pictures are different views of a universe
given by Borromean axes with order four symmetry (everything is repeated
four times on your walk). The fourth picture shows axes of order 20 symmetry.
The Borromean axes have become very small and very distant. Each cluster
point has 20 beams incident on it, but not all of them are visible. In the
fifth and sixth pictures, we have two different views of the situation when
the Borromean axes have been pushed all the way to infinity, and are not,
therefore, in the picture. So we have a hyperbolic structure on the complement
of the Borromean axes.

The pictures given here are insufficient to understand the proof in
full – for that you need to study the video, available from Jones and Bartlett,
20 Park Plaza, Boston, Ma 02116, US. It is possible to take the video’s
proof by pictures and convert it, step by step, into a formal proof in the
classical style.