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In mathematics, an nsphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an nsphere of radius r is defined as the set of points in (n + 1)dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. In symbols:
It is an ndimensional manifold in Euclidean (n + 1)space. In particular, a 0sphere is a pair of points on a line, a 1sphere is a circle in the plane, and a 2sphere is an ordinary sphere in threedimensional space. Spheres of dimension n > 2 are sometimes called hyperspheres, with 3spheres sometimes known as glomes. The nsphere of unit radius centered at the origin is called the unit nsphere, denoted S^{n}. The unit nsphere is often referred to as the nsphere. An nsphere is the surface or boundary of an (n + 1)dimensional ball, and is an ndimensional manifold. For n ≥ 2, the nspheres are the simply connected ndimensional manifolds of constant, positive curvature. The nspheres admit several other topological descriptions: for example, they can be constructed by gluing two ndimensional Euclidean spaces together, by identifying the boundary of an ncube with a point, or (inductively) by forming the suspension of an (n − 1)sphere.
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