Hypersphere
A hypersphere is a higher-dimensional analogue of a sphere. A hypersphere of radius r in n-dimensional Euclidean space consists of all points at distance r from a given fixed point (the centre of the hypersphere).The "volume" it encloses is
The "surface area" of this hypersphere is
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Terminology
Confusingly, geometers and topologists have different names for the above object in n-dimensions.Geometrical terminology
Following Coxeter, geometers call the above object an n-sphere. Hence, an ordinary sphere in three dimensions would be called a "3-sphere".Topological terminology
Topologists call the above object an (n−1)-manifold or an (n−1)-sphere. Hence, the special case of an ordinary sphere in three dimensions would be called a "2-sphere".