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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 center. The surface of the hypersphere is a manifold of one dimension less than the ambient space. As the radius increases the curvature of the hypersphere decreases; in the limit a hypersphere approaches the zero curvature of a hyperplane. Both hyperplanes and hyperspheres are hypersurfaces.

The term hypersphere was introduced by Duncan Sommerville in his discussion of models for non-Euclidean geometry.[1] The first one mentioned is a 3-sphere in four dimensions.

Some spheres are not hyperspheres: suppose S is a sphere in Em where m < n and the space had 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.

References

D. M. Y. Sommerville (1914) The Elements of Non-Euclidean Geometry, page 193, link from University of Michigan Historical Math Collection

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Hypersphere