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Sphere
Sphere
A sphere is defined as the locus of the points in three dimensions that are equidistant from a particular point called the center. Note that the center of a sphere is unique.
It is generally assumed that the sphere is embedded in real-valued space (
) unless otherwise stated.
The equation for a sphere centered at the origin is
where 
is the length
of the radius.
A unit sphere is a sphere with radius 1.
The formula for the volume
of a sphere with radius 
is
The formula for the surface area
of a sphere with radius 
is
Generalization
A sphere can be generalized to 
dimensions. For 
, a generalized sphere is called a hypersphere (when no value of 
is given, one can generally assume that “hypersphere” means 
). In the same manner, the definitions
of center, radius, and unit sphere can also be generalized to 
dimensions.
The formula for an 
-dimensional sphere is
where 
is the length of the radius. Note that when 
, the formula reduces to the formula for a circle, so a circle is a 2-dimensional “sphere”. A one dimensional sphere is a pair of points (filled-in, it would be a line)!
The volume of an 
-dimensional sphere with radius 
is
where 
is the gamma function. Curiously, for any fixed
, the volume of the 
-d sphere approaches zero as 
approaches infinity. Contrast this to the volume of an 
-d box, which always has a volume in proportion
to 
(with 
the side
length of the box) which increases without bound
when 
. Note that, for any positive
integer
and for any radius 
,
. Also note that the volume of the 
-d unit sphere 
has a maximum precisely at 
.
To illustrate how to use the formula for 
and to provide some evidence of the claims made about 
, the values 
, 
, and 
will be calculated here.
Topological Treatment
In topology
and other contexts, spheres are treated slightly differently. Let the 
-sphere be the set
where
can be any norm, usually the Euclidean norm. Notice that 
is defined here as a subset
of
.
Thus, 
is two points on the real
line:
{$-1$}
\rput[t](1,0){$1$}
\end{pspicture}](http://myyn.org/static/assets/mathbrowser/article/2620/images/img46.png){kind=small}
is the unit circle:
{$x$}
\rput[l](0,1.2){$y$}
\pscircle(0,0){1}
\end{pspicture}](http://myyn.org/static/assets/mathbrowser/article/2620/images/img48.png)
is the unit sphere in the everyday sense of the word. It might seem like a strange naming convention to say, for instance, that the 
-sphere is in three-dimensional space. The explanation is that 
refers to the sphere's “intrinsic” dimension as a manifold, not the dimension to whatever space in which it happens to be immersed.
Sometimes this definition is generalized even more. In topology we usually fail to distinguish homeomorphic
spaces, so all homeomorphic images
of 
into any topological space are also called 
. It is usually clear
from context whether 
denotes the specific unit sphere in
or some arbitrary homeomorphic image.