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Graphs of volumes (V) and surface areas (S) of n-spheres of radius 1. In the SVG file, hover over a point to see its decimal value.

In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center. It is a manifold of codimension one, i.e. with one dimension less than that of 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. Hyperplanes and hyperspheres are examples of 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: if S is a sphere in Em where m < n 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.


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

Further reading

  • Kazuyuki Enomoto (2013) Review of an article in International Electronic Journal of Geometry.MR3125833
  • Jemal Guven (2013) "Confining spheres in hyperspheres", Journal of Physics A 46:135201, doi:10.1088/1751-8113/46/13/135201
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