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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).
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 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.
A sphere (from Greek σφαῖρα, sphaîra) [1] is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. [2] That given point is the center of the sphere, and r is the sphere's radius.
An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a disk.
The lens space L(1,0) is the 3-sphere, and the lens space L(2,1) is 3 dimensional real projective space. Lens spaces can be represented as Seifert fiber spaces in many ways, usually as fiber spaces over the 2-sphere with at most two exceptional fibers, though the lens space with fundamental group of order 4 also has a representation as a ...
In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...