Twisting and Turning in Space

ion and classification continues, What's Euclidean On a perfectly flat surface, parallel lines never meet. A triangle drawn on such a plane has interior angles that add up to 1800. A straight line is the shortest distance between two points. These and other postulates and results are part of the geometry first invented by Euclid more than 2,000 years ago. Later mathematicians refined and extended his ideas to higher dimensions, and the postulates became part of what is now called Euclidean geometry. However, surfaces exist on which the Euclidean rules do not hold. On the surface of a sphere, for example, the shortest distance between two points is an arc (called a great circle) following the sphere's curvature. A triangle bounded by "straight" lines has angles that add up to more than 1800. What appear to be parallel lines intersect in antipodal points. Because the geometry on a sphere is different from that on a flat surface, it is one example of a non-Euclidean geometry. Manifolds are surfaces and shapes, sometimes very complex, that appear to be Euclidean when a small region is examined, but on a large scale fail to follow the rules for a Euclidean geometry A sphere is an example of a manifold because a small region of the sphere appears to be flat, although on a larger scale the surface is clearly curved. I. Peterson 42 SCIENCE NEWS, VOL. 122 This content downloaded on Thu, 14 Mar 2013 16:50:16 PM All use subject to JSTOR Terms and Conditions the domain of discourse has been enormously extended into higher dimensions, into topological (rubber-sheet geometric) models, into curves, surfaces and diverse shapes called manifolds (see box, p. 42). Roughly speaking, our discovery of new geometric objects has generally exceeded our ability to classify them. (It's as if Darwin, rather than being a passenger on the earth-bound Beagle, had been a passenger on Starship Enterprise, discovering on each new world exotic forms that defied easy classification.) Now, at long last, mathematicians' efforts to classify geometric shapes may be approaching an end. One of the major unresolved sections of this multidimensional puzzle was just completed, and the outlines of the remaining pieces are beginning to take shape. The origins of this scheme, however, stretch back to the turn of the century. Geometry's Darwin was the incomparable French mathematician Henri Poincare (1854-1912), the one person who almost beat Einstein to the theory of relativity. In studies conducted around 1900, Poincare explored many special surfaces and volumes as potential domains for the solutions to differential equations. Solutions of some differential equations can be readily represented by paths taken in ordinary twoor three-dimensional Euclidean space (see box, p. 42). Other equations are best modeled by solutions on a sphere, a torus (doughnut-shape), or even more exotic objects -such as a twisted loop with only one side, known as a Mobius strip (see p. 36); or a three-dimensional tube, known as a Klein bottle, whose inside surface loops back on itself to merge with its outside. Poincare set out to classify these many forms and surfaces, and in so doing helped create the modern field of topology. In geometry, notions such as distance and angle are of paramount importance. But what counts in topology is whether one shape can be continuously deformed into another. All five Platonic solids, for example, are topologically equivalent not only to each other, but also to a sphere. However, because it has a hole in it, a torus is quite a different topological object. Poincare discovered an algebraic way to detect and study the pattern of holes in a topological surface. His strategy in this "algebraization" of topology was to examine the behavior of simple closed paths (those which, like a circle, begin and end at the same point) on the surface. All closed paths on a sphere, for instance, can be continuously shrunk to a single point, whereas only some of those drawn on a torus can be shrunk to a point. That's because on a torus there are paths of fundamentally different species: Those winding through the center hole are quite different from those winding around the center hole. And both are different from the cyclic paths that loop through the center as they go around the torus. Poincare showed that species of paths on surfaces form a group a simple, algebraic concept that represents geometric symmetry. And the nature of a fundamental Poincare group tells a lot about the nature of its corresponding surface. This representation of topological surfaces by Poincare groups is now part of the theory of homotopy (literally "of the same shape") -so named because it represents the mathematical transformation of one curve into another, or the shrinking of a curve into a point (when that is possible). Once the exclusive tool of algebraic topologists, homotopies have recently come to be employed by graphic artists and computer scientists who wish to provide smooth videotape transitions between frames of artistic animation. Poincare's fundamental homotopy group is sufficiently discriminating to identify all two-dimensional surfaces: Two such surfaces are topologically equivalent if (and only if) they correspond to the same Poincare group. In the third dimension, however, things turn out to be less simple. In fact, Poincare's effort to extend homotopy classification to threedimensional objects ended where it began -with the simplest case, the sphere. Poincare conjectured that three-dimensional spheres behave just like two-dimensional ones that every three-dimensional object that had the same homotopy groups as the sphere is topologically equivalent to a sphere. In other words, there should be no fake spheres: An object behaving like a sphere should be a sphere. Poincare's conjecture and its generalization to higher dimensions ranks as one of the most challenging unsolved problems in mathematics. And it's worth noting that whole fields of mathematics have developed in the course of testing, on a case-by-case basis, the Poincare conjecture. Topologists quickly discovered as they explored geometric structures in higher dimensions that neither the intuition nor the vocabulary of simple geometry would suffice. So they introduced the "manifold" as a general term for certain topological objects that (in any dimension) include a great variety of surfaces and spaces. The great task of topology in this century has been to classify manifolds to discover the origin of these exotic geometric species. The first major advance came in 1954, when Poincare's countryman, Rene Thom, discovered an important clue to the organization of manifolds: When two manifolds together serve as a boundary for a third, they share important related characteristics. Thom's theory, called "cobordism," provided insight into ways manifolds behave in dimensions greater than four. In 1962, Stephen Smale of the University of California at Berkeley used an extension of cobordism techniques to prove the Poincare conjecture for manifolds of the fifth and higher dimensions. Shortly after, John R. Stallings and E. Christopher Zeeman provided alternate proofs of the same results. (Ten years later Thom and Zeeman used ideas derived from this work on classification of manifolds to investigate and classify "elementary catastrophes" a subject that spawned widespread interest and controversy when it was applied to research in the behavioral sciences [SN: 4/2/77, p. 218].) Manifolds in higher dimensions (those above four) are in many respects paradoxical: Their classification proved easier C)