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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}{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:
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\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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