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{\small

\begin{center}
{\Large \textbf{HOW TO DISTINGUISH BETWEEN}}

\smallskip

{\Large \textbf{PLANE SYMMETRIES}}

\bigskip

{\Large Recognition Chart for Plane Periodic Patterns}
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\begin{tabular}{lllrrll}
&& \multicolumn{1}{c}{Highest}    & \multicolumn{1}{c}{Non-trivial}
&& \multicolumn{1}{c}{Order}      & \multicolumn{1}{c}{Glide}
&  \multicolumn{1}{c}{Generating} & \multicolumn{1}{c}{Distinguishing} \\
\multicolumn{1}{c}{Type}       & \multicolumn{1}{c}{Lattice}
&  \multicolumn{1}{c}{Rotation}   & \multicolumn{1}{c}{Reflections}
&  \multicolumn{1}{c}{Region}     & \multicolumn{1}{c}{Properties} \\
\hline
p1   & parallelogram & 1 &  no &  no & 1 unit    \\
p2   & parallelogram & 2 &  no &  no & 1/2 unit  \\
pm   & parallelogram & 1 & yes &  no & 1/2 unit  \\
pg   & rectangular   & 1 &  no & yes & 1/2 unit  \\
cm   & rhombic       & 1 & yes & yes & 1/2 unit  \\
pmm  & rectangular   & 2 & yes &  no & 1/4 unit  \\
pmg  & rectangular   & 2 & yes & yes & 1/4 unit  & parallel \\
&               &   &     &     &           & reflection axes \\
pgg  & rectangular   & 2 &  no & yes & 1/4 unit  \\
cmm  & rhombic       & 2 & yes & yes & 1/4 unit  & perpendicular \\
&               &   &     &     &           & reflection axes \\
p4   & square        & 4 &  no &  no & 1/4 unit  \\
p4m  & square        & 4 & yes & yes & 1/4 unit  & 4-fold centres \\
&               &   &     &     &           & on reflection axes \\
p4g  & square        & 4 & yes & yes & 1/4 unit  & 4-fold centres
\textit{not} \\
&               &   &     &     &           & on reflection axes \\
p3   & hexagonal     & 3 &  no &  no & 1/3 unit  \\
p3m1 & hexagonal     & 3 & yes & yes & 1/6 unit  & all 3-fold centres \\
&               &   &     &     &           & on reflection axes \\
p31m & hexagonal     & 3 & yes & yes & 1/6 unit  & \textit{not} all
3-fold centres \\
&               &   &     &     &           & on reflection axes \\
p6   & hexagonal     & 6 &  no &  no & 1/6 unit  \\
p6m  & hexagonal     & 6 & yes & yes & 1/12 unit \\
\end{tabular}
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\noindent
Notes:
\renewcommand{\labelenumi}{(\theenumi)}
\begin{enumerate}
\item A rotation through an angle of $360^{\circ}/n$ is said to have
order $n$.  A glide-reflection is non-trivial if its component
translation and reflection are not symmetries of the pattern.
\item A smallest region of the plane having the property that the set
of its images under the translation group covers the plane is called
a unit of the pattern.
\item A generating region is a smallest region whose images under the
full symmetry group of the pattern cover the plane.
\end{enumerate}

}

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