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 s 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 I ve been interested in the Millennium problems since I first read about them several years ago It was exciting to read about the first one to be solved I never took topology in college, though, so I have to admit that much of this went right over my head If you wanted to know without reading all the math, yes, the Poincare conjecture turned out to be true Pretty cool stuff