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.
\nA
sphere is, roughly speaking, a
ball-shaped object. In
mathematics, a sphere comprises only the
surface of the ball, and is therefore
hollow.\nIn 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
-
\nThe points on the sphere with radius
r and center at the origin can be parametrized via\n:\n:\n:\n(see
trigonometric functions and
spherical coordinates).
A sphere of any radius centered at the origin is described by the following
differential equation:\n:\nThis 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:
\n:\nand its enclosed
volume is:
\n:
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.
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.
\n* a 2-sphere is therefore an ordinary sphere\n* a 1-sphere is a circle\n* 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.
\n* a 3-sphere is therefore an ordinary sphere\n* a 2-sphere is a circle\n* a 1-sphere is a pair of points
See also
External link
\n*Mathworld website
\nCategory:Differential geometry\nCategory:Differential topology\nCategory:Surfaces
\n\n\n\n\n\n\n\n\nsimple:Sphere\n\n
