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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.


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

Further reading

Kazuyuki Enomoto (2013) Review of an article in International Electronic Journal of Geometry.MR 3125833
Jemal Guven (2013) "Confining spheres in hyperspheres", Journal of Physics A 46:135201, doi:10.1088/1751-8113/46/13/135201

Mathematics Encyclopedia

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