sphereSphere2001-10-15 17:59:592008-03-08 21:52:18Definitioncenterradiusunit spherehyperspheren-sphere$n$-sphere\usepackage{amssymb, amsmath, amsthm, alltt, setspace, pstricks, pst-plot}
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\newcommand{\floor}[1]{\lfloor{#1}\rfloor}
\newcommand{\ceil}[1]{\lceil{#1}\rceil}\section{Sphere}
A \emph{sphere} is defined as the locus of the points in three dimensions that are equidistant from a particular point called the \emph{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 \PMlinkescapetext{\emph{radius}}.
A \emph{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
\[ A = 4 \pi r^2. \]
\section{Generalization}
A sphere can be generalized to $n$ dimensions. For $n > 3$, a generalized sphere is called a \emph{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.
\begin{center}
$\begin{array}{lll|lll|ll}
V(4,1) & = \displaystyle \frac{\pi^{\frac{4}{2}}1^4}{\Gamma(\frac{4}{2}+1)} & & V(5,1) & = \displaystyle \frac{\pi^{\frac{5}{2}}1^5}{\Gamma(\frac{5}{2}+1)} & & V(6,1) & = \displaystyle \frac{\pi^{\frac{6}{2}}1^6}{\Gamma(\frac{6}{2}+1)} \\
& & & & & & & \\
& = \displaystyle \frac{\pi^2}{\Gamma(3)} & & & = \displaystyle \frac{\pi^2 \sqrt{\pi}}{\Gamma(\frac{7}{2})} & & & = \displaystyle \frac{\pi^3}{\Gamma(4)} \\
& & & & & & & \\
& = \displaystyle \frac{\pi^2}{2} & & & = \displaystyle \frac{\pi^2 \sqrt{\pi}}{\frac{15}{8} \sqrt{\pi}} & & & = \displaystyle \frac{\pi^3}{6} \\
& & & & & & & \\
& \approx 4.9348 & & & = \displaystyle \frac{8\pi^2}{15} & & & \approx 5.1677 \\
& & & & & & & \\
& & & & \approx 5.2638 & & & \end{array}$
\end{center}
\section{Topological Treatment}
In topology and other contexts, spheres are treated slightly differently. Let the $n$-\emph{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:
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$S^1$ is the unit circle:
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$S^2$ is the unit sphere in the everyday sense of the \PMlinkescapetext{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 \PMlinkescapetext{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.