% LaTeX source for How to distinguish between plane symmetries
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\begin{document}
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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}
\end{center}
\begin{center}
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\begin{tabular}{lllrrll}
&& \multicolumn{1}{c}{Highest} & \multicolumn{1}{c}{Non-trivial}
&& \multicolumn{1}{c}{Helpful} \\
&& \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}
\end{center}
\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}
}
\end{document}
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