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 ($\mathbb{R}^3$ ) unless otherwise stated.

The equation for a sphere centered at the origin is $$ x^2+y^2+z^2=r^2 $$

where $r$ 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 $r$ is $$ V = \frac{4}{3} \pi r^3. $$

The formula for the surface area of a sphere with radius $r$ is $$ A = 4 \pi r^2. $$

Generalization

A sphere can be generalized to $n$ dimensions. For $n > 3$ , a generalized sphere is called a hypersphere (when no value of $n$ is given, one can generally assume that ``hypersphere'' means $n = 4$ ). In the same manner, the definitions of center, radius, and unit sphere can also be generalized to $n$ dimensions.

The formula for an $n$ -dimensional sphere is $$ {x_1}^2 + {x_2}^2 + \dots + {x_n}^2 = r^2 $$

where $r$ is the length of the radius. Note that when $n=2$ , 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 $n$ -dimensional sphere with radius $r$ is $$ V(n,r) = \frac{\pi^{\frac{n}{2}}r^n}{\Gamma(\frac{n}{2}+1)} $$

where $\Gamma(n)$ is the gamma function. Curiously, for any fixed $r$ , the volume of the $n$ -d sphere approaches zero as $n$ approaches infinity. Contrast this to the volume of an $n$ -d box, which always has a volume in proportion to $s^n$ (with $s$ the side length of the box) which increases without bound when $s \ge 1$ . Note that, for any positive integer $n$ and for any radius $r$ , $V(n,r)=V(n,1)r^n$ . Also note that the volume of the $n$ -d unit sphere $V(n,1)$ has a maximum precisely at $n=5$ .

To illustrate how to use the formula for $V(n,r)$ and to provide some evidence of the claims made about $V(n,r)$ , the values $V(4,1)$ , $V(5,1)$ , and $V(6,1)$ will be calculated here.

Topological Treatment

In topology and other contexts, spheres are treated slightly differently. Let the $n$ -sphere be the set $$ S^n = \{ x \in \RR^{n+1} : ||x|| = 1 \} $$

where $|| \cdot ||$ can be any norm, usually the Euclidean norm. Notice that $S^n$ is defined here as a subset of $\RR^{n+1}$ .

Thus, $S^0$ is two points on the real line:

$S^1$ is the unit circle:

$S^2$ 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 $2$ -sphere is in three-dimensional space. The explanation is that $2$ 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 $S^n$ into any topological space are also called $S^n$ . It is usually clear from context whether $S^n$ denotes the specific unit sphere in $\RR^{n+1}$ or some arbitrary homeomorphic image.