# Hypersphere

For spheres in hyperspace, see n-sphere.

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 E 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.

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Hypersphere.