In Hyperbolic Space, Size Matters

摘要

Mathematicians since the 19th century have explored the strange realm of hyperbolic geometry, in which parallel lines behave in ways that Euclid never imagined. And cos-mologists have pondered its implications for the universe ever since Einstein introduced curved space in his general theory ofrnrelativity. Modern researchers have long known that among the peculiarities of hyperbolic geometry, there is a hyperbolic three-dimensional space, or 3-manifold, of least volume. They've even long had a candidate for the smallest hyperbolic space, a tiny snarl known as the Weeks manifold. What they didn't have was proof the theory couldn't cough up something smaller yet. Now they have that too.