Poincare Conjecture
Table of Contents


In a 1904 paper, the French mathematician Jules Henri Poincaré stated that a sphere is a sphere is a sphere. You can punch, kick and throw it; you can inflate or deflate it; you can mold the sphere into another shape. But in the world of topology, no matter what you do to it, the resulting deformed, twisted and complicated form is still a sphere.

Also, you cannot poke a hole in it. You cannot, for example, turn your sphere into a donut. You cannot turn it into a coffee cup with a handle, frames for your eyeglasses or a key ring. You can stick your finger into it, but you can't actually puncture the surface or reach inside. If you break the surface in any way you've ventured into a different genus of topological objects.

Say you're walking down a street, and you encounter a strange and complicated shape whose surface sports peaks and valleys, mountains and molehills, but no holes. If you were a mathematician, you may want to study the way that functions behave on it. Poincaré's conjecture says that no matter what it looks like, it's a sphere. The conjecture gives mathematicians a short and easy way to identify a deformed blob as a sphere in disguise.

In its original form, however, the Poincaré conjecture concerned three-dimensional spheres (i.e., 3-spheres). These shapes are difficult, perhaps impossible, to visualize—the universe, for instance, is thought of as a 3-sphere. Even without being able to picture it, draw it or know that it exists, we can do math on a 3-sphere. We can calculate distances between points. Any system that can be characterized by three numbers automatically determines a three-dimensional shape. In baseball, for example, if you tally the numbers of runs, pitches and fouls for each inning of a game that doesn't go into extra innings, you have established nine data points in a three-dimensional space. With those nine points, you can make statements about the "shape" you have created.




