N-sphere

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

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).

Considered extrinsically, as a hypersurface embedded in ⁠${\displaystyle (n+1)}$⁠-dimensional Euclidean space, an ⁠${\displaystyle n}$⁠-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 ⁠${\displaystyle (n+1)}$⁠-dimensional ball. In particular:

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

Given a Cartesian coordinate system, the unit ⁠${\displaystyle n}$⁠-sphere of radius ⁠${\displaystyle 1}$⁠ can be defined as:

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

Considered intrinsically, when ⁠${\displaystyle n\geq 1}$⁠, the ⁠${\displaystyle n}$⁠-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the ⁠${\displaystyle n}$⁠-sphere are called great circles.

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

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

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

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

Cartesian coordinates

The set of points in ⁠${\displaystyle (n+1)}$⁠-space, ⁠${\displaystyle (x_{1},x_{2},\ldots ,x_{n+1})}$⁠, that define an ⁠${\displaystyle n}$⁠-sphere, ⁠${\displaystyle S^{n}(r)}$⁠, is represented by the equation:

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

where ⁠${\displaystyle \mathbf {c} =(c_{1},c_{2},\ldots ,c_{n+1})}$⁠ is a center point, and ⁠${\displaystyle r}$⁠ is the radius.

The above ⁠${\displaystyle n}$⁠-sphere exists in ⁠${\displaystyle (n+1)}$⁠-dimensional Euclidean space and is an example of an ⁠${\displaystyle n}$⁠-manifold. The volume form ⁠${\displaystyle \omega }$⁠ of an ⁠${\displaystyle n}$⁠-sphere of radius ⁠${\displaystyle r}$⁠ is given by

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

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

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

n-ball

Main article: Ball (mathematics)

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

Specifically:

• A ⁠${\displaystyle 1}$⁠-ball, a line segment, is the interior of a 0-sphere.
• A ⁠${\displaystyle 2}$⁠-ball, a disk, is the interior of a circle (⁠${\displaystyle 1}$⁠-sphere).
• A ⁠${\displaystyle 3}$⁠-ball, an ordinary ball, is the interior of a sphere (⁠${\displaystyle 2}$⁠-sphere).
• A ⁠${\displaystyle 4}$⁠-ball is the interior of a 3-sphere, etc.

Topological description

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

See also: Volume of an n-ball and Unit sphere § Volume and area

Let ⁠${\displaystyle S_{n-1}}$⁠ be the surface area of the unit ⁠${\displaystyle (n-1)}$⁠-sphere of radius ⁠${\displaystyle 1}$⁠ embedded in ⁠${\displaystyle n}$⁠-dimensional Euclidean space, and let ⁠${\displaystyle V_{n}}$⁠ be the volume of its interior, the unit ⁠${\displaystyle n}$⁠-ball. The surface area of an arbitrary ⁠${\displaystyle (n-1)}$⁠-sphere is proportional to the ⁠${\displaystyle (n-1)}$⁠st power of the radius, and the volume of an arbitrary ⁠${\displaystyle n}$⁠-ball is proportional to the ⁠${\displaystyle n}$⁠th power of the radius.

Graphs of volumes (⁠${\displaystyle V_{n}}$⁠) and surface areas (⁠${\displaystyle S_{n-1}}$⁠) of n-balls of radius 1.

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

${\displaystyle V_{0}=1.}$

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

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

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

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

The interior of a 2-sphere in three-dimensional space is the unit ⁠${\displaystyle 3}$⁠-ball.

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

In general, ⁠${\displaystyle S_{n-1}}$⁠ and ⁠${\displaystyle V_{n}}$⁠ are given in closed form by the expressions

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

where ⁠${\displaystyle \Gamma }$⁠ is the gamma function.

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

Recurrences

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

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

Equivalently, representing the unit ⁠${\displaystyle n}$⁠-ball as a union of concentric ⁠${\displaystyle (n-1)}$⁠-sphere shells,

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

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

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

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

We may define a coordinate system in an ⁠${\displaystyle n}$⁠-dimensional Euclidean space which is analogous to the spherical coordinate system defined for ⁠${\displaystyle 3}$⁠-dimensional Euclidean space, in which the coordinates consist of a radial coordinate ⁠${\displaystyle r}$⁠, and ⁠${\displaystyle n-1}$⁠ angular coordinates ⁠${\displaystyle \varphi _{1},\varphi _{2},\ldots ,\varphi _{n-1}}$⁠, where the angles ⁠${\displaystyle \varphi _{1},\varphi _{2},\ldots ,\varphi _{n-2}}$⁠ range over ⁠${\displaystyle [0,\pi ]}$⁠ radians (or ⁠${\displaystyle [0,180]}$⁠ degrees) and ⁠${\displaystyle \varphi _{n-1}}$⁠ ranges over ⁠${\displaystyle [0,2\pi )}$⁠ radians (or ⁠${\displaystyle [0,360)}$⁠ degrees). If ⁠${\displaystyle x_{i}}$⁠ are the Cartesian coordinates, then we may compute ⁠${\displaystyle x_{1},\ldots ,x_{n}}$⁠ from ⁠${\displaystyle r,\varphi _{1},\ldots ,\varphi _{n-1}}$⁠ with:^[3][a]