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Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point.
The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables r {\displaystyle r} is the radius, C = 2 π r {\displaystyle C=2\pi r} is the circumference (the length of any one of its great circles ),
In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. In the intrinsic approach, a great circle is a geodesic; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane ...
The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space. In topology, the n-sphere is an example of a compact topological manifold without boundary. A topological sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere).
The two-dimensional analogue of the Poincaré conjecture says that any two-dimensional topological manifold which is closed and connected but non-homeomorphic to the two-dimensional sphere must possess a loop which cannot be continuously contracted to a point. (This is illustrated by the example of the torus, as above.)
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.
In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere.
The 3-sphere is an especially important 3-manifold because of the now-proven Poincaré conjecture. Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold).