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The Poincaré conjecture is this:  the set of points at an equal radius from the center in four-dimensional space (known as a "three-dimensional sphere") is the only finite, bounded, simply connected object that [[manifold|locally looks like three-dimensional space]] and has the property that all loops and spheres within it can be shrunk to points.

The Poincaré conjecture is this:  the set of points at an equal radius from the center in four-dimensional space (known as a "three-dimensional sphere") is the only finite, bounded, simply connected object that [[manifold|locally looks like three-dimensional space]] and has the property that all loops and spheres within it can be shrunk to points.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.

Put another way, the conjecture states that if strings, surfaces, and other geometric objects "loop around" a three-dimensional space in the same way they loop around what mathematicians call a "three sphere" (see below), then the space in question is a three sphere itself.

===The three sphere===

===The three sphere===