An n-ball of radius 1 (a “unit ball”) will just fit inside an n-cube with sides of length 2. The surface of the ball kisses the center of each face of the cube. In this configuration, what fraction of the cubic volume is filled by the ball? The question is answered easily in the familiar low-dimensional spaces we are all accustomed to living in. At the bottom of the hierarchy is onedimensional geometry, which is rather dull: Everything looks like a line segment. A 1-ball with r = 1 and a 1-cube with s = 2 are actually the same object— a line segment of length 2. Thus in one dimension the ball completely fills the cube; the volume ratio is 1.0. In two dimensions, a 2-ball inside a 2-cube is a disk inscribed in a square, and so this problem can be solved with one of my childhood formulas. With r = 1, the area πr2 is simply π, whereas the area of the square, s2 , is 4; the ratio of these quantities is about 0.79. In three dimensions, the ball’s volume is 4 ∕3π, whereas the cube has a volume of 8; this works out to a ratio of approximately 0.52. On the basis of these three data points, it appears that the ball fills a smaller and smaller fraction of the cube as n increases. There’s a simple, intuitive argument suggesting that the trend will continue: The regions of the cube that are left vacant by the ball are the corners. Each time n increases by 1, the number of corners doubles, so we can expect ever more volume to migrate into the nooks and crannies near the cube’s vertices. Beyond the fifth dimension, the volume of a unit n-ball decreases as n increases! I tried a few larger values of n, finding that V(20,1) is about 0.0258, and V(100,1) is in the neighborhood of 10–40. Thus it looked very much like the n-ball dwindles away to nothing as n approaches infinity.
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Exotic spheres:
In 1956, John Milnor was investigating 7-dimensional manifolds when he found a shape which seemed very strange. On one hand, it contained no holes, and so it seemed to be a sphere. On the other hand, the way it was curved around was not like a sphere at all. Initially Milnor thought that he had found a counterexample to the 7-dimensional version of the Poincaré conjecture: a shape with no holes, which was not a sphere. But on closer inspection, his new shape could morph into a sphere (as Poincaré insists it must be able to do), but - remarkably - it could not do so smoothly. So, although it was topologically a sphere, in differential terms it was not.
Milnor had found the first exotic sphere, and he went on to find several more in other dimensions. In each case, the result was topologically spherical, but not differentially so. Another way to say the same thing is that the exotic spheres represent ways to impose unusual notions of distance and curvature on the ordinary sphere.
In dimensions one, two, and three, there are no exotic spheres, just the usual ones. This is because the topological and differential viewpoints do not diverge in these familiar spaces. Similarly in dimensions five and six there are only the ordinary spheres, but in dimension seven, suddenly there are 28. In higher dimensions the number flickers around between 1 and arbitrarily large numbers:
Dimension123456789101112131415161718Number of spheres111?11282869921321625621616
The realm which remains the most mysterious, even today, is 4-dimensional space. No exotic spheres have yet been found here. At the same time no-one has managed to prove that none can exist. The assertion that there are no exotic spheres in four dimensions is known as the smooth Poincaré conjecture. In case anyone has got this far and is still not sure, let me make this clear: the smooth Poincaré conjecture is not the same thing as the Poincaré conjecture! Among other differences, the Poincaré conjecture has been proved, but the smooth Poincaré conjecture remains stubbornly open today.
Further reading
Geometry of higher dimensional space
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Dr. Stefan Gruenwald