Hyperspheres

Hypersphere is the generalization of a sphere to higher dimensions. An n-dimensional hypersphere, denoted S^n(r), is the set of all points in (n+1)-dimensional Euclidean space R^{n+1} at distance r from the origin. It is the boundary of the (n+1)-dimensional ball B^{n+1}(r). For example, S^1 is a circle, and S^2 is the ordinary surface of a 3D sphere.

Two standard quantities associated with S^n(r) are its surface area and the volume of the enclosing ball. The surface area is A_n(r) = 2 π^{(n+1)/2} r^n / Γ((n+1)/2). The volume of the (n+1)-dimensional ball is V_{n+1}(r) = π^{(n+1)/2} r^{n+1} / Γ((n+3)/2). The unit-sphere area is A_n = A_n(1). These satisfy the relation d/dr V_{n+1}(r) = A_n(r).

The formulas involve the gamma function Γ(z), which generalizes factorials (Γ(k+1) = k! for integers k). In particular, A_n(1) = 2 π^{(n+1)/2} / Γ((n+1)/2). The expressions yield familiar results in low dimensions: S^1 has circumference 2πr, and S^2 has area 4πr^2.

Properties commonly noted include that the hypersphere has constant positive curvature and high symmetry, with isometry group O(n+1). In high dimensions, the volume of the unit ball and the surface area behave non-intuitively, often shrinking rapidly as n grows. Hyperspheres play a central role in geometry, analysis, and applications involving high-dimensional spaces.