n-sphere

In mathematics, an n-sphere or hypersphere is an n-dimensional generalization of the 1-dimensional circle and 2-dimensional sphere to any non-negative integer n. The n-sphere is the setting for n-dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in (n + 1)-dimensional Euclidean space, an n-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an (n + 1)-dimensional ball. In particular:

• The 0-sphere is the pair of points at the ends of a line segment (1-ball).
• The 1-sphere is a circle, the circumference of a disk (2-ball) in the two-dimensional plane.
• The 2-sphere, often simply called a sphere, is the boundary of a 3-ball in three-dimensional space.
• The 3-sphere is the boundary of a 4-ball in four-dimensional space.
• The (n – 1)-sphere is the boundary of an n-ball.

Given a Cartesian coordinate system, the unit n-sphere of radius 1 can be defined as:

${\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\}.}$

Considered intrinsically, when n ≥ 1, the n-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the n-sphere are called great circles.

The stereographic projection maps the n-sphere onto n-space with a single adjoined point at infinity; under the metric thereby defined, ${\displaystyle \mathbb {R} ^{n}\cup \{\infty \}}$ is a model for the n-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit n-sphere is called an n-sphere. Under inverse stereographic projection, the n-sphere is the one-point compactification of n-space. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere. When n ≥ 2 it is simply connected; the 1-sphere (circle) is not simply connected; the 0-sphere is not even connected, consisting of two discrete points.