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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:
- A = 4πr2
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.
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
See also
External link
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