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n-sphere
Generalized sphere of dimension n (mathematics) / From Wikipedia, the free encyclopedia
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In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the
-dimensional circle and
-dimensional sphere to any non-negative integer
. The
-sphere is the setting for
-dimensional spherical geometry.
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Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an
-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an
-dimensional ball. In particular:
- The
-sphere is the pair of points at the ends of a line segment (
-ball).
- The
-sphere is a circle, the circumference of a disk (
-ball) in the two-dimensional plane.
- The
-sphere, often simply called a sphere, is the boundary of a
-ball in three-dimensional space.
- The 3-sphere is the boundary of a
-ball in four-dimensional space.
- The
-sphere is the boundary of an
-ball.
Given a Cartesian coordinate system, the unit -sphere of radius
can be defined as:
Considered intrinsically, when , the
-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the
-sphere are called great circles.
The stereographic projection maps the -sphere onto
-space with a single adjoined point at infinity; under the metric thereby defined,
is a model for the
-sphere.
In the more general setting of topology, any topological space that is homeomorphic to the unit -sphere is called an
-sphere. Under inverse stereographic projection, the
-sphere is the one-point compactification of
-space. The
-spheres admit several other topological descriptions: for example, they can be constructed by gluing two
-dimensional spaces together, by identifying the boundary of an
-cube with a point, or (inductively) by forming the suspension of an
-sphere. When
it is simply connected; the
-sphere (circle) is not simply connected; the
-sphere is not even connected, consisting of two discrete points.