The Poincaré Conjecture.

As you probably noticed, I’ve got a new design here on the dish, one that was long overdue. I’d like to thank (and credit) Thomas Griffin for designing and setting up the theme, and reader Sara Showalter for designing that awesome custom header image.

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The Poincaré 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 of Prime Obsession, is another one of the seven.)

In his 2007 book, The Poincaré Conjecture: In Search of the Shape of the Universe (still on sale for $6.38 as a bargain book on amazon), Donal O’Shea, Dean of Faculty at Mt. Holyoke College and a professor of mathematics, gives a brisk history of the Conjecture with a quick mention of its solution. The first half of the book, from Euclid and Pythagoras up to Henri Poincaré and the early 20th century, was relatively fast-moving (for a math book) and easy to follow, but when O’Shea got deeper into topological discussions of the Conjecture, his explanations became shorter and I found myself getting lost.

The Poincaré Conjecture states that:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

In lay terms – and I apologize if I get this wrong – 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 “3-sphere.” 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 – 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.

Henri Poincaré, 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 “simply connected” – 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 – and “closed” – meaning if you walked on its surface, you would never reach an edge or boundary because the space closes around on itself – 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.

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. Poincaré 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.

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 – and I mean really understand them – 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.

I’m also not sure that O’Shea managed to deliver on the book’s subtitle. That the Poincaré Conjecture’s answer might 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 Poincaré dodecahedral space (also called a Poincaré 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.

If you’re interested in these great problems of mathematics, I’d recommend John Derbyshire’s Prime Obsession, 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, also rather tricky to follow because of its heavy use of topology but with a bit more drama to help the reader plow through the less scrutable parts.

Next up: Sticking with math, I’m halfway through Alex Bellos’ Here’s Looking at Euclid, 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.