Sphere





A sphere is, roughly speaking, a ball-shaped object. In mathematics, a
sphere is a quadric consisting only of a surface and is therefore hollow. In
non-mathematical usage a sphere is often considered to be solid;
mathematicians call this the interior of the sphere.

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.

In coordinate geometry a sphere with centre (x0,Êy0,Êz0) and radius r is the
set of all points (x,y,z) such that

     (x - x0)2 + (y - y0)2 + (z - z0)2 = r2

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

     x = r cos(φ) sin(θ)
     y = r sin(φ) sin(θ) ÊÊÊÊÊ (0 ≤ θ < π and -π <
     φ ≤ π)
     z = r cos(θ)

The surface area of a sphere of radius r is 4πr2, and its volume is
4πr3/3. 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 water drops (in the absence of gravity) 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. 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.

Spheres can be generalized to other 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. A 2-sphere is therefore an ordinary sphere, while a
1-sphere is a circle and a 0-sphere is a pair of points. The n-sphere of
unit radius centered at the origin is denoted Sn and is often referred to as
"the" n-sphere.

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

----------------------------------------------------------------------------

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.