over the surface of the unit -ball. This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the unit cube is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed. In seventy dimensions, less than of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out. Uniformly at random within the n-ball With a point selected uniformly at random from the surface of the unit -sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit -ball. If is a number generated uniformly at random from the interval and is a point selected uniformly at random from the unit -sphere, then is uniformly distributed within the unit -ball. Alternatively, points may be sampled uniformly from within the unit -ball by a reduction from the unit -sphere. In particular, if is a point selected uniformly from the unit -sphere, then is uniformly distributed within the unit -ball (i.e., by simply discarding two coordinates). If is sufficiently large, most of the volume of the -ball will be contained in the region very close to its surface, so a point selected from that volume will also probably be close to the surface. This is one of the phenomena leading to the so-called curse of dimensionality that arises in some numerical and other applications. Specific spheres 0-sphere The pair of points with the discrete topology for some . The only sphere that is not path-connected. Parallelizable. 1-sphere Commonly called a circle. Has a nontrivial fundamental group. Abelian Lie group structure ; the circle group. Homeomorphic to the real projective line. 2-sphere Commonly simply called a sphere. For its complex structure, see Riemann sphere. Equivalent to the complex projective line 3-sphere Parallelizable, principal -bundle over the 2-sphere, Lie group structure . 4-sphere Equivalent to the quaternionic projective line, . . 5-sphere Principal -bundle over . . It is undecidable whether a given -dimensional manifold is homeomorphic to for . 6-sphere Possesses an almost complex structure coming from the set of pure unit octonions. . The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. 7-sphere Topological quasigroup structure as the set