Thursday, January 28, 2010

Life in four dimensions

By Steve Rensberry

Russian mathematician Grigori Perelman's proof of the famous Poincare conjecture in 2002-03 and the subsequent confirmation in 2006 of his efforts drew an enormous amount of interest - and rightly so considering that it has been an entire century since Henri Poincare first made his famous conjecture.

In simple terms the conjecture asserts that spherical objects defined in terms of three dimensional Euclidian space have the same type of surface connectivity as do spherical objects defined in terms of four dimensional Euclidian space.

Stephen Ornes in the Aug. 26, 2006 edition of Seed Magazine referred to it as something which "gives mathematicians a short and easy way to identify a deformed blob as a sphere in disguise."

I was among those intrigued by it, but what I found hard to get my head around beyond just the terminology and concepts was envisioning precisely what kind of shape a three-dimensional sphere would have in four dimensions, or what anything in four dimensions would look like for that matter. We're talking about dimensions in Euclidian space mind you and not that dealing with space-time construction, or Minkowski space.

But visualizing things in four dimensions is a nearly impossible feat. Even in our familiar three-dimensional way of looking at things we don't really see things in three dimensions. We don't see both sides, the inside and every angle of an object all at the same time. What we see is the two-dimensional surfaces of three-dimensional objects.

In three dimensions things are defined using three pairs of cardinal directions, represented by altitude, latitude and longitude. In four dimensions there is an addition set of cardinal directions which are orthogonal (at right angles) to each of the others.

Furthermore, in Euclidian geometry a two-dimensional sphere is defined by a set of three different points in three-dimensional space and referred to as a 2-sphere or 2-manifold. A three-dimensional sphere is defined by a set of four different points in four-dimensional space and referred to as a 3-sphere or 3-manifold.

As Ornes explains: "When most people think of a sphere, they generally consider the space that a sphere occupies—a ping-pong ball, for example. When topologists talk about a sphere, they are talking exclusively about its surface."

What Poincare suggested and Perelman proved was that all three-dimensional, finite, simply-connected manifolds which do not have holes are spheres.

A coffee cup with a handle is not a sphere, neither is a doughnut, a car tire or a pair of pants. A flattened saucer? A dinner plate? A square box which doubles as a coffee table? Well, those indeed are spheres for the simple reason that their surface, in what form it can be transformed into without ripping it apart, is a closed, simply connected, three-dimensional manifold.

Perelman's solution involved performing a type of mathematical surgery on the singularities or sections of a three-dimensional sphere which are malformed or crinkled, essentially creating two separate, topologically identical spheres - and lending another proof of William Thurston's geometrization conjecture in the meantime. It was a solution to the problematic areas in manipulated three-manifolds that no one had thought of before.

As with a lot of things, the importance of the conjecture isn't so much the idea that it would help us understand the universe itself - which some believe may be a 3-sphere - but the simple way in which Perelman arrived at his solution.

As Dennis Overbye of the New York Times wrote in an Aug. 18, 2006 article: "Everybody agrees that it is no surprise that the conjecture is true. What is surprising is the way it was proved, using mathematics far removed from traditional topology, establishing links no one had suspected between disparate fields and techniques.

Hmmmm. Isn't that the way it always is?