Hypersphere

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.

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

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.

References

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

Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hypersphere Hyperrectangle Demihypercube Cross-polytope Simplex
Dimensions by number	Zero One Two Three Four Five Six Seven Eight Nine n-dimensions Negative dimensions
Category

Hypersphere

References

Further reading