\section{Isometries}

\begin{remark}
	It is convenient to begin our discussion with the definition of an isometry (from the Greek \textit{isometros}, meaning "equal measure") in $\R^n$.
\end{remark}

\begin{definition}[Isometry]
	An \textit{isometry} of $n$-dimensional space $\R^n$ is a function from $\R^n$ onto $\R^n$ that preserves distance.
\end{definition}

\begin{definition}[Symmetry Group of a Figure in $\mathbf{\R^n}$]
	Let $F$ be a set of points in $\R^n$. the \textit{symmetry group of $F$} in $\R^n$ is the set of all isometries of $\R^n$ that carry $F$ onto itself. The group operation is function composition.
\end{definition}