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The 3-sphere is homeomorphic to the one-point compactification of R 3. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere. The homology groups of the 3-sphere are as follows: H 0 (S 3, Z) and H 3 (S 3, Z) are both infinite cyclic, while H i (S 3, Z) = {} for all other indices i.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
The 3-sphere is the boundary of a -ball in four-dimensional space. The ( n − 1 ) {\displaystyle (n-1)} -sphere is the boundary of an n {\displaystyle n} -ball. Given a Cartesian coordinate system , the unit n {\displaystyle n} -sphere of radius 1 {\displaystyle 1} can be defined as:
So, in effect, Hamilton showed a special case of the Poincaré conjecture: if a compact simply-connected 3-manifold supports a Riemannian metric of positive Ricci curvature, then it must be diffeomorphic to the 3-sphere. If, instead, one only has an arbitrary Riemannian metric, the Ricci flow equations must lead to more complicated singularities.
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space R 4. If a point has coordinates, P ( x , y , z , w ) , then x 2 + y 2 + z 2 + w 2 = 1 characterizes those points on the unit 3-sphere centered at the origin.
The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space. Alternatively, the point ( z 0 , z 1 ) can be mapped to the ratio z 1 / z 0 in the Riemann sphere C ∞ .
2 Physics. 3 Formulae yielding ... where SV is the surface volume of a 3-sphere and r is the radius. Regular convex polygons ... More formulas of this nature can be ...
S 2: a 2-sphere is an ordinary sphere; S 3: a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional ...