—<\/p>\n
The Poincar\u00e9 Conjecture was one of seven Millennium Prize Problems identified by the Clay Institute in 2000 as the most significant unsolved problems (or unproven theorems) in mathematics, and at this point it is the only one of the seven problems that has been solved. Such a solution should have earned its developer, in this case a somewhat reclusive Russian named Grigori Perelman, a million-dollar prize, but Perelman rejected the prize and the Fields Medal he was to be awarded for his solution. (The Riemann Hypothesis, which I discussed in my review last year<\/a> of Prime Obsession<\/i>, is another one of the seven.) <\/p>\n In his 2007 book, The Poincar\u00e9 Conjecture: In Search of the Shape of the Universe<\/a> The Poincar\u00e9 Conjecture states that:<\/p>\n Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.<\/p><\/blockquote>\n In lay terms \u2013 and I apologize if I get this wrong \u2013 it means that any four-dimensional shape that is internally continuous and has no boundary can be mapped, point for point, to the four-dimensional shape called the \u201c3-sphere.\u201d The 3-sphere contains every point in 4-space equidistant from a single center; a point in 4-space is defined the set of coordinates (w, x, y, z). Think of a three-dimensional sphere, defined by all points (x, y, z) 1 unit distant from a single point, such as (0, 0, 0); this sphere will include (1, 0, 0), (0, -1, 0), (0, ?2, ?2), and all other points such that the square root of their sums equals one. (This is similar to the Pythagorean Theorem, but with another variable added to the sum.) We can picture this sphere in 3-space, so while we can’t picture the 3-sphere in 4-space, we can at least follow the math \u2013 the 3-sphere of unit 1 and center (0, 0, 0, 0) will include the points (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and so on.<\/p>\n Henri Poincar\u00e9, a prolific and brilliant French mathematician who built on work done by Bernhard Riemann, conjectured but could not prove that any four-dimensional shape that is \u201csimply connected\u201d – where any loop including two points can be reduced to a single point, meaning there is no disruption in the overall shape inside of such a loop \u2013 and \u201cclosed\u201d – meaning if you walked on its surface, you would never reach an edge or boundary because the space closes around on itself \u2013 can me mapped, point for point, to the 3-sphere. As it turns out, this conjecture was extremely hard to prove, requiring mathematial concepts that did not exist at the time of the conjecture, and relevant to the question of the shape of the universe.<\/p>\n O’Shea did a solid job going into the history of first Euclidean and then non-Euclidean geometry, with interesting digressions on the lasting nature of the mathematical works of the ancient Greeks, how discoveries by Arab and Indian mathematicians (who were often religious leaders as well) spread to Europe, and how much knowledge was lost along the way, including much of Euclid’s work lost in the fire at the library of Alexandria. Poincar\u00e9 himself is not an ideal central figure for a work of non-fiction, only jumping off the page in the chapter outlining his rivalry (and flame war, in letters) with the Prussian nationalist mathematician Felix Klein. <\/p>\n Where O’Shea lost me was with very brief introductions of critical terms used to describe the search for the Conjecture’s proof, then repeated use of those terms without sufficient explanation. I never encountered tensors in any of the math classes I took in school, and I don’t know what Ricci flows are (they were only created\/discovered in 1981), or Betti numbers, or Laplace operators, but you need to understand those terms \u2013 and I mean really understand them \u2013 to follow the descriptions of the various steps leading up to and including Perelman’s solution. This is no small task; I’m asking O’Shea to describe upper-level college mathematics topics to readers who may not have gone beyond first-order calculus in a way that they will understand it. I don’t think he achieved that goal here.<\/p>\n I’m also not sure that O’Shea managed to deliver on the book’s subtitle. That the Poincar\u00e9 Conjecture’s answer might<\/i> help us understand the shape of the universe does not appear to be in any doubt. That it pushes us further toward understanding the shape of the universe is unclear, both from the book and from what I could find online that didn’t exceed my understanding. There does seem to be some thought that the universe might be a Poincar\u00e9 dodecahedral space (also called a Poincar\u00e9 homology sphere), a closed 3-manifold that is not simply connected, formed by taking opposing faces of a dodecahedron, rotating one to align with its opposite, and then smushing the dodecahedron and gluing each pair of faces together to form a 3-manifold in 4-space that is not homeomorphic to the 3-sphere. And I’ll stop there before I get further out of my league.<\/p>\n If you’re interested in these great problems of mathematics, I’d recommend John Derbyshire’s Prime Obsession<\/i>, which I mentioned above and found more accessible than O’Shea’s book even though the problem under consideration, the Riemann Hypothesis, remains unsolved and likely has no practical application. O’Shea’s book reminded me of Amir Aczel’s slim volume called Fermat’s Last Theorem<\/a>![\"\"](\"http:\/\/www.assoc-amazon.com\/e\/ir?t=meadowpartyco-20&l=as2&o=1&a=B001PTG4IC&camp=217145&creative=399373\")
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<\/i>, sent to me by the publisher earlier this year. It’s a fun tour of mathematical puzzles and oddities with a few dashes of number theory thrown in, but nothing you couldn’t follow if you have a high school degree.<\/p>\n","protected":false},"excerpt":{"rendered":"