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Sphere

A sphere is, roughly speaking, a ball-shaped object. In mathematics, a sphere comprises only the surface of the ball, and is therefore hollow. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball).

More precisely, a sphere is the set of points in 3-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.

Equations

In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the set of all points (x,y,z) such that

<math>(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2 \,<math>

A solid sphere with center (x₀, y₀, z₀) and radius r is the set of all points (x,y,z) such that

<math>(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 \le r^2 \,<math>

The points on the sphere with radius r can be parametrized via

<math> x = x_0 + r \sin \theta \; \cos \phi <math>

<math> y = y_0 + r \sin \theta \; \sin \phi \qquad (0 < \theta < \pi \mbox{ and } -\pi < \phi < \pi) \,<math>

<math> z = z_0 + r \cos \theta \,<math>

(see also trigonometric functions and spherical coordinates).

A sphere of any radius centered at the origin is described by the following differential equation:

<math> x \, dx + y \, dy + z \, dz = 0. <math>

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other.

The surface area of a sphere of radius r is:

<math>A = 4 \pi r^2 \,<math>

and its enclosed volume is:

<math>V = \frac{4 \pi r^3}{3}<math>

The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are spheres, because the surface tension tries to minimize surface area.

The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes.

A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.

Generalization to higher dimensions

Spheres can be generalized to higher dimensions. For any natural number n, an n-sphere is the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number. Here, the choice of number reflects the dimension of the sphere as a manifold.

• a 0-sphere is a pair of points
• a 1-sphere is a circle
• a 2-sphere is an ordinary sphere
• a 3-sphere is a sphere in 4-dimensional Euclidean space

Spheres for n ≥ 3 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sⁿ and is often referred to as "the" n-sphere. The notation Sⁿ is also often used to denote any set with a given structure (topological space, topological manifold, smooth manifold, etc.) identical (homeomorphic, diffeomorphic, etc.) to the structure of Sⁿ above.

An n-sphere is an example of a compact n-manifold.

See also

• Riemann sphere
• 3-sphere, hypersphere
• ball
• torus