In mathematics, an n-sphere or hypersphere is an n {\displaystyle n} -dimensional generalization of the 1 {\displaystyle 1} -dimensional circle and 2 {\displaystyle 2} -dimensional sphere to any non-negative integer n {\displaystyle n} . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general n {\displaystyle n} -sphere is embedded in an n + 1 {\displaystyle n+1} -dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension n ≥ 3 {\displaystyle n\geq 3} which are thus embedded in a space of dimension n + 1 ≥ 4 {\displaystyle n+1\geq 4} , which means that they cannot be easily visualized. The n {\displaystyle n} -sphere is the setting for n {\displaystyle n} -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in ( n + 1 ) {\displaystyle (n+1)} -dimensional Euclidean space, an n {\displaystyle 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 ) {\displaystyle (n+1)} -dimensional ball. In particular:
The 0 {\displaystyle 0} -sphere is the pair of points at the ends of a line segment ( 1 {\displaystyle 1} -ball). The 1 {\displaystyle 1} -sphere is a circle, the circumference of a disk ( 2 {\displaystyle 2} -ball) in the two-dimensional plane. The 2 {\displaystyle 2} -sphere, often simply called a sphere, is the boundary of a 3 {\displaystyle 3} -ball in three-dimensional space. The 3-sphere is the boundary of a 4 {\displaystyle 4} -ball in four-dimensional space. The ( n − 1 ) {\displaystyle (n-1)} -sphere is the boundary of an n {\displaystyle n} -ball. Given a Cartesian coordinate system, the unit n {\displaystyle n} -sphere of radius 1 {\displaystyle 1} can be defined as:
S n = { x ∈ R n + 1 : ‖ x ‖ = 1 } . {\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\}.}
Considered intrinsically, when n ≥ 1 {\displaystyle n\geq 1} , the n {\displaystyle n} -sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the n {\displaystyle n} -sphere are called great circles. The stereographic projection maps the n {\displaystyle n} -sphere onto n {\displaystyle n} -space with a single adjoined point at infinity; under the metric thereby defined, R n ∪ { ∞ } {\displaystyle \mathbb {R} ^{n}\cup \{\infty \}} is a model for the n {\displaystyle n} -sphere. In the more general setting of topology, any topological space that is homeomorphic to the unit n {\displaystyle n} -sphere is called an n {\displaystyle n} -sphere. Under inverse stereographic projection, the n {\displaystyle n} -sphere is the one-point compactification of n {\displaystyle n} -space. The n {\displaystyle n} -spheres admit several other topological descriptions: for example, they can be constructed by gluing two n {\displaystyle n} -dimensional spaces together, by identifying the boundary of an n {\displaystyle n} -cube with a point, or (inductively) by forming the suspension of an ( n − 1 ) {\displaystyle (n-1)} -sphere. When n ≥ 2 {\displaystyle n\geq 2} it is simply connected; the 1 {\displaystyle 1} -sphere (circle) is not simply connected; the 0 {\displaystyle 0} -sphere is not even connected, consisting of two discrete points.
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