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
- <math>\pi^{n/2}r^n\over\Gamma(n/2+1)<math>
where Γ is the gamma function.
The "surface area" of this hypersphere is
- <math>2\pi^{n/2}r^{n-1}\over\Gamma(n/2)<math>
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".
See also
