tanik
2次元結晶空間の中の対称操作で，1点を不動の特異点にする対称操作の作る群は10種類．
$$1, m, 2, 2mm, 3, 3m, 4, 4mm, 6, 6mm$$
-------------------------------------
$$2mm=2⊗m$$
$$3m=3⊙m$$
$$4mm=4⊙m$$
$$6mm=6⊙m$$
$$6=3⊗2$$
$$4=2○4(mod2)$$
--------------------------------
$$6mm\left\{ \begin{array}{@{\,} c @{\, } }
\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 6\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 3 \\[0mm]
\vartriangleright 2
\end{array} \right. \\[0mm]
\supset m
\end{array} \right. \\[0mm]
\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 3m\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 3 \\[0mm]
\supset m
\end{array} \right. \\[0mm]
\supset 2
\end{array} \right. \\[0mm]
\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 2mm\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 2 \\[0mm]
\vartriangleright m
\end{array} \right. \\[0mm]
\supset 3
\end{array} \right.
\end{array} \right. $$
$$4mm\left\{ \begin{array}{@{\,} c @{\, } }
\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 4\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 2 \\[0mm]
○4(mod2)
\end{array} \right. \\[0mm]
\supset m
\end{array} \right. \\[0mm]
\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 2mm\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 2 \\[0mm]
\vartriangleright m
\end{array} \right. \\[0mm]
\supset 2
\end{array} \right.
\end{array} \right. $$
$$3m\left\{ \begin{array}{@{\,} c @{\, } }
\vartriangleright 3 \\[0mm]
\supset m
\end{array} \right. $$
■対称群を系統的に見る
$$\begin{array}{|c|c|c|c|c|}
\hline
\begin{array}{@{\,} c @{\, } }
次元数 \to \\[0mm]
周期軸数 \downarrow
\end{array} & 0 & 1 & 2 & 3 \\[0mm]
\hline
0 & G_{0,0}=1 & G_{1,0}=2 & G_{2,0}=10 & G_{3,0}=32 \\[0mm]
\hline
1 & \times & G_{1,1}=2 & G_{2,1}=7 & G_{3,1}=75 \\[0mm]
\hline
2 & \times & \times & G_{2,2}=17 & G_{3,2}=80 \\[0mm]
\hline
3 & \times & \times & \times & G_{3,3}=230 \\[0mm]
\hline
\end{array}$$
■2次元のブラベー格子
２次元のバラベー格子は5種類
