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It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold.
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).
This is the set of points in 3-dimensional Euclidean space found exactly one unit away from the origin. It is called the 2-sphere, S 2, for reasons given below. The same idea applies for any dimension n; the equation x 2 0 + x 2 1 + ⋯ + x 2 n = 1 produces the n-sphere as a geometric object in (n + 1)-dimensional space. For example, the 1 ...
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.
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 complement to any connected knot or graph in a 3-dimensional sphere is of type (,); this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos. [ 1 ] Any compact , connected, non-positively curved manifold M is a K ( Γ , 1 ) {\displaystyle K(\Gamma ,1)} , where Γ = π 1 ( M ) {\displaystyle \Gamma =\pi ...
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 ...
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 ...
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