Sphere

Sphere Books was a British paperback publisher of the 1960s - 1980s.

Sphere is the name of a book written by Michael Crichton, which was subsequently turned into a movie by the same name.

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.

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

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

The points on the sphere with radius r and center at the origin can be parametrized via

(see trigonometric functions and spherical coordinates).

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

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:

and its enclosed volume is:

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.

Table of contents

1 Generalisation to n-dimensions

1.1 Topological definition
1.2 Geometrical definition

2 See also
3 External link

Generalisation to n-dimensions

Spheres can be generalized to higher dimensions. Confusingly, there are two conventions for a definition in use — firstly, the most common definition, adopted by topologists and differential geometers; and secondly, a definition used by certain other geometers.

Topological definition

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 2-sphere is therefore an ordinary sphere
a 1-sphere is a circle
a 0-sphere is a pair of points

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.

Geometrical definition

For any natural number n, an n-sphere is the set of points in n-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 number of coordinates needed to express the equation defining the sphere.

a 3-sphere is therefore an ordinary sphere
a 2-sphere is a circle
a 1-sphere is a pair of points

External link

Mathworld website