\section{The Crystallographic Groups}

\begin{remark}
	The seven frieze groups catalog all symmetry groups that leave a design invariant under all multiples of just one translation. However, there are 17 additional kinds of discrete plane symmetry groups that arise from infinitely repeating designs in a plane. these groups are the symmetry groups of plane patterns whose subgroups of translations are isomorphic to $\Z \oplus \Z$. Consequently, the patterns are invariant under linear combinations of two linearly independent translations. These 16 groups were first studied by the 19th-century crystallographers and often called the \textit{plane crystallographic groups}. Another term occasionally used for these groups is \textit{wallpaper groups}.
\end{remark}