I recently read The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea. This 2007 book was one of the flood that greeted the proof, by Grigori Perelman, of the Poincaré conjecture. It is probably the best of them.
The Poincaré conjecture relates to the 3-sphere, the 3-dimensional object sitting in 4-dimensional space defined by x2 + y2 + z2 + t2 = R2. As the diagram below illustrates, this can be seen as the union of an infinite set of hollow balloons, strung along the t axis, each defined by x2 + y2 + z2 = R2 − t2
The 3-sphere x2 + y2 + z2 + t2 = R2 (horizontal t scale exaggerated 10 times, and only showing 21 of the infinitely many hollow balloons).
These hollow balloons can be nested inside each other to make two solid balls (one sequence from the left to the centre, and one sequence from the right to the centre). A 3-sphere is therefore equivalent to a pair of solid balls, where the two surfaces of the balls are imagined to coincide. That is, any path “leaving” one ball immediately “enters” the other at the corresponding point. As a consequence, the 3-sphere has no boundary.
As an example, consider a path from the centre of the left sphere, moving upwards. On reaching the north pole of the left sphere, the path immediately shifts to the north pole of the right sphere, but moving downwards. Then, on reaching the south pole of the right sphere, the path immediately shifts to the south pole of the left sphere, now moving upwards again. In this way, the path eventually reaches its starting point.
The 3-sphere x2 + y2 + z2 + t2 = R2 as a pair of solid balls, where the two surfaces of the balls are imagined to coincide (thus leaving the 3-sphere without a boundary).
The Poincaré conjecture – now proved by Perelman – concerns potential universes that are connected, and in small regions look like Euclidean 3-space, but are also finite in size, and lacking a boundary. Henri Poincaré conjectured that the only such potential universe with the additional property that each loop in the space could be continuously tightened to a point was the 3-sphere.
Both the road to that conjecture, and the road from the conjecture to the proof, make for fascinating reading. So, if you haven’t already read this book, look for a copy.
As a side issue, O’Shea also touches on the interesting suggestion that the universe as described in Dante’s Paradiso (see image above) is a 3-sphere. It is still a possibility that our universe has that shape too.