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paulbrown

So – the shape of the universe. It’s a giant ball, right? Especially when you think of its beginning in a big bang. But that brings up the awkward question of what’s outside the ball. Space (universe) is not infinite. It’s believed to be finite, but without a boundary. It becomes easier to understand this if you consider two-dimensional beings living in a spherical (the two-dimensional surface of a ball) universe. Their universe is finite, but has no boundaries. There are no edges, and if they start off from one point and keep going in the same direction they’ll come back to where they started. Our universe is finite and without boundary in the same way. If you get on a spaceship and keep going in the same direction, eventually you’ll be back in the same neighborhood! This one is harder to imagine, isn’t it? In the case of two-dimensional people living on a sphere, we can see how it can be finite but without boundary because we can see how the sphere bends in a third dimension. But how is it for our three-dimensional universe? There’s no fourth dimension to bend in. Reading this book didn’t make it any easier for me to really understand how the universe can be finite but without a boundary. All I can do is quote the two-dimensional analogy, but I’m still a three-dimensional earthling. But even assuming that the universe is finite and without boundary – is it a three-sphere? To go back to the two-sphere analogy, just because Magellan sailed in the same direction and came back to where he started doesn’t mean that the earth is a sphere. It can also be doughnut shaped, and the same would still happen. No one really knows what the shape of the universe is. There’s a lot of evidence for it being flat (whatever that means). And the Poincare Conjecture: It says that a finite, no-boundary space that is “simply connected” is a three-sphere. This question is obviously of great interest both to mathematicians and to the physicists studying the geometry of the universe. (We still don’t know if the universe is “simply connected” or not. A ball is simply connect, but something like a doughnut is not simply connected.) Unlike Reimann’s Hypothesis, the Poincare Conjecture was finally proved after much heartbreak and agony – by an eccentric Russian mathematician named Gregori Perelman who didn’t even accept the award for it. The book tells the story of the conjecture and the man who proved it. Good pop-science and math history.

2022-10-21 05:29