# n-sphere

## Generalized sphere of dimension n (mathematics) / From Wikipedia, the free encyclopedia

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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:

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

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, $\mathbb {R} ^{n}\cup \{\infty \}$ 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.

For any natural number *n*, an *n*-sphere of radius *r* is defined as the set of points in (*n* + 1)-dimensional Euclidean space that are at distance *r* from some fixed point **c**, where *r* may be any positive real number and where **c** may be any point in (*n* + 1)-dimensional space. In particular:

- a 0-sphere is a pair of points {
*c*−*r*,*c*+*r*}, and is the boundary of a line segment (1-ball). - a 1-sphere is a circle of radius
*r*centered at**c**, and is the boundary of a disk (2-ball). - a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
- a 3-sphere is a 3-dimensional sphere in 4-dimensional Euclidean space.

### Cartesian coordinates

The set of points in (*n* + 1)-space, (*x*_{1}, *x*_{2}, ..., *x*_{n+1}), that define an *n*-sphere, *S*^{n}(*r*), is represented by the equation:

- $r^{2}=\sum _{i=1}^{n+1}(x_{i}-c_{i})^{2},$

where **c** = (*c*_{1}, *c*_{2}, ..., *c*_{n+1}) is a center point, and *r* is the radius.

The above *n*-sphere exists in (*n* + 1)-dimensional Euclidean space and is an example of an *n*-manifold. The volume form *ω* of an *n*-sphere of radius *r* is given by

- $\omega ={\frac {1}{r}}\sum _{j=1}^{n+1}(-1)^{j-1}x_{j}\,dx_{1}\wedge \cdots \wedge dx_{j-1}\wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1}={\star }dr$

where ${\star }$ is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case *r* = 1. As a result,

- $dr\wedge \omega =dx_{1}\wedge \cdots \wedge dx_{n+1}.$

*n*-ball

The space enclosed by an *n*-sphere is called an (*n* + 1)-ball. An (*n* + 1)-ball is closed if it includes the *n*-sphere, and it is open if it does not include the *n*-sphere.

Specifically:

### Topological description

Topologically, an *n*-sphere can be constructed as a one-point compactification of *n*-dimensional Euclidean space. Briefly, the *n*-sphere can be described as *S*^{n} = ℝ^{n} ∪ {∞}, which is *n*-dimensional Euclidean space plus a single point representing infinity in all directions.
In particular, if a single point is removed from an *n*-sphere, it becomes homeomorphic to ℝ^{n}. This forms the basis for stereographic projection.^{[1]}

Let *S*_{n−1} be the surface area of the unit (*n* − 1)-sphere of radius 1 embedded in n-dimensional Euclidean space, and let *V*_{n} be the volume of its interior, the unit n-ball. The surface area of an arbitrary (*n* − 1)-sphere is proportional to the (*n* − 1)st power of the radius, and the volume of an arbitrary n-ball is proportional to the nth power of the radius.

The 0-ball is sometimes defined as a single point. The 0-dimensional Hausdorff measure is the number of points in a set. So

- $V_{0}=1.$

A unit 1-ball is a line segment whose points have a single coordinate in the interval [−1, 1] of length 2, and the 0-sphere consists of its two end-points, with coordinate {−1, 1}.

- $S_{0}=2,\quad V_{1}=2.$

A unit 1-sphere is the unit circle in the Euclidean plane, and its interior is the unit disk (2-ball).

- $S_{1}=2\pi ,\quad V_{2}=\pi .$

The interior of a 2-sphere in three-dimensional space is the unit 3-ball.

- $S_{2}=4\pi ,\quad V_{3}={\tfrac {4}{3}}\pi .$

In general, *S*_{n−1} and *V*_{n} are given in closed form by the expressions

- $S_{n-1}={\frac {2\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}{\bigr )}}},\quad V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\frac {n}{2}}+1{\bigr )}}}$

where Γ is the gamma function.

As n tends to infinity, the volume of the unit n-ball (ratio between the volume of an n-ball of radius 1 and an n-cube of side length 1) tends to zero.^{[2]}

### Recurrences

The *surface area*, or properly the *n*-dimensional volume, of the *n*-sphere at the boundary of the (*n* + 1)-ball of radius *R* is related to the volume of the ball by the differential equation

- $S_{n}R^{n}={\frac {dV_{n+1}R^{n+1}}{dR}}={(n+1)V_{n+1}R^{n}}.$

Equivalently, representing the unit *n*-ball as a union of concentric (*n* − 1)-sphere *shells*,

- $V_{n+1}=\int _{0}^{1}S_{n}r^{n}\,dr={\frac {1}{n+1}}S_{n}.$

We can also represent the unit (*n* + 2)-sphere as a union of products of a circle (1-sphere) with an *n*-sphere. Then $S_{n+2}=2\pi V_{n+1}.$ Since *S*_{1} = 2π *V*_{0}, the equation

- $S_{n+1}=2\pi V_{n}$

holds for all *n*. Along with the base cases $S_{0}=2,$ $V_{1}=2$ from above, these recurrences can be used to compute the surface area of any sphere or volume of any ball.