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A hypersphere is a generalization of the circle and sphere to any number of dimensions. The n-sphere or comprises the set of points in (n + 1)-dimensional Euclidean space that are all a fixed nonzero distance from a given point, which is designated as its center. The fixed distance is the radius, and the resulting shape is an n-dimensional manifold.

The n-sphere is the boundary of the closed (n + 1)-ball, which is the set of points in whose distance from the center is less than or equal to a given radius. The open (n + 1)-ball is the closed (n + 1)-ball without its boundary.

Examples[edit | edit source]

Excluding the point, the hyperspheres up to 10D are the following:

Orthoplexes by dimension
Rank Name Picture Rank Name Picture
1 Dyad
6 Hexasphere
2 Circle 7 Heptasphere
3 Sphere 8 Octasphere
4 3-sphere (glome) 9 Enneasphere
5 Pentasphere 10 Decasphere