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In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its centre. It is a manifold of codimension one--that is, with one dimension less than that of the ambient space.
As the hypersphere's radius increases, its curvature decreases. In the limit, a hypersphere approaches the zero curvature of a hyperplane. Hyperplanes and hyperspheres are examples of hypersurfaces.
The term hypersphere was introduced by Duncan Sommerville in his 1914 discussion of models for non-Euclidean geometry. The first one mentioned is a 3-sphere in four dimensions.
Some spheres are not hyperspheres: If S is a sphere in Em where , and the space has n dimensions, then S is not a hypersphere. Similarly, any n-sphere in a proper flat is not a hypersphere. For example, a circle is not a hypersphere in three-dimensional space, but it is a hypersphere in the plane.