Hypersphere

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 centre. It is a manifold of codimension one—that is, with one dimension less than that of the ambient space.

As the hypershere'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 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

↑ Sommerville, D. M. Y. (1914). "'Space Curvature' and the Philosophical Bearing of Non-Euclidean Geometry" (PDF). In Milne, William P. The Elements of Non-Euclidean Geometry. Bell's Mathematical Series for Schools and Colleges. London: G. Bell and Sons. p. 193 – via University of Michigan Historical Math Collection.

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