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.^{[1]} The first one mentioned is a 3-sphere in four dimensions.

Some spheres are not hyperspheres: If *S* is a sphere in E^{m} 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.

**^**Duncan Sommerville (1914) Elements of Non-Euclidean Geometry, page 193 via Internet Archive

- 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