N-sphere

2-sphere wireframe as an orthogonal projection
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue), and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect ⟨0,0,0,1⟩ have an infinite radius (= straight line).

In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they exist as shapes in 1- and 2-dimensional space. As such, the -sphere is the setting for -dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -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 -dimensional ball. In particular:

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

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

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

The stereographic projection maps the -sphere onto -space with a single adjoined point at infinity; under the metric thereby defined, is a model for the -sphere.

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