Merge branch 'master' into master

12 files changed, 926 insertions, 0 deletions

diff --git a/buch/papers/punktgruppen/tikz/atoms-grid-force.tex b/buch/papers/punktgruppen/tikz/atoms-grid-force.tex
new file mode 100644
index 0000000..05742cf
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/atoms-grid-force.tex
@@ -0,0 +1,42 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex, 
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = black, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] {
+      \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\
+      \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+      \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\
+    };
+
+    \foreach \d in {NW, N, NE} {
+      \draw[orange, very thick, <-] (\d) to ++(0,.7);
+    }
+
+    \foreach \d in {SW, S, SE} {
+      \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+    }
+
+    \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/atoms-grid-still.tex b/buch/papers/punktgruppen/tikz/atoms-grid-still.tex
new file mode 100644
index 0000000..4e43856
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/atoms-grid-still.tex
@@ -0,0 +1,33 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex, 
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = black, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] {
+      \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\
+      \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+      \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\
+    };
+    \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex
new file mode 100644
index 0000000..e4c3f93
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex
@@ -0,0 +1,47 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex, 
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = black, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
+    \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
+    \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
+    \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
+    \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
+    \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
+
+    \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+    % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+    \draw[orange, very thick, <-] (C6) to ++(.7,0);
+    \draw[orange, very thick, <-] (C3) to ++(-.7,0);
+
+    \node[black] (E) {\(\vec{E}_p\)};
+    \begin{scope}[node distance = .5mm]
+      \node[blue!50, right = of E] {\(-\)};
+      \node[red!50, left = of E] {\(+\)};
+    \end{scope}
+    % \draw[gray, thick, dotted] (E) to ++(0,2);
+    % \draw[gray, thick, dotted] (E) to ++(0,-2);
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex
new file mode 100644
index 0000000..892ab42
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex
@@ -0,0 +1,52 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex, 
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = black, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
+    \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
+    \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
+    \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
+    \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
+    \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
+
+    \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+    % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+    \foreach \d in {C1, C2} {
+      \draw[orange, very thick, <-] (\d) to ++(0,.7);
+    }
+
+    \foreach \d in {C4, C5} {
+      \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+    }
+
+    \node[black] (E) {\(\vec{E}_p\)};
+    \begin{scope}[node distance = .5mm]
+      \node[red!50, right = of E] {\(+\)};
+      \node[blue!50, left = of E] {\(-\)};
+    \end{scope}
+    % \draw[gray, thick, dotted] (E) to ++(0,2);
+    % \draw[gray, thick, dotted] (E) to ++(0,-2);
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex
new file mode 100644
index 0000000..2eb78ba
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex
@@ -0,0 +1,34 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex, 
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = black, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} {
+      \node[charge, \t] (C\i) at (\x:1.5cm) {};
+    }
+
+    \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+    \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {};
+    % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/combine-symmetries.tex b/buch/papers/punktgruppen/tikz/combine-symmetries.tex
new file mode 100644
index 0000000..fa669ae
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/combine-symmetries.tex
@@ -0,0 +1,57 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+\begin{tikzpicture}[
+    >=latex, 
+    dot/.style = {
+      draw, circle, thick, black, fill = gray!40!white,
+      minimum size = 2mm,
+      inner sep = 0pt,
+      outer sep = 1mm,
+    },
+  ]
+
+  \node[dot] (A1) at (0,0) {};
+  \node[below left] at (A1) {\(A\)};
+
+  \node[dot] (A2) at (2.5,0) {};
+  \node[below right] at (A2) {\(A'\)};
+
+  \draw[red!80!black, thick, ->] 
+    (A1) to node[midway, below] {\(\vec{Q}\)} (A2);
+
+  \node[dot] (B1) at (120:2.5) {};
+  \node[above left] at (B1) {\(B\)};
+
+  \draw[green!70!black, thick, ->]
+    (A1) ++(.5,0) arc (0:120:.5)
+    node[midway, above, xshift=1mm] {\(C_n\)};
+  \draw[red!80!black, dashed, thick, ->] (A1) to (B1);
+
+  \node[dot] (B2) at ($(A2)+(60:2.5)$) {};
+  \node[above right] at (B2) {\(B'\)};
+
+  \draw[green!70!black, thick, dashed, ->]
+    (A2) ++(-.5,0) arc (180:60:.5);
+  \draw[red!80!black, dashed, thick, ->] (A2) to (B2);
+
+  \draw[cyan!40!blue, thick, ->]
+    (B1) to node[above, midway] {\(\vec{Q}'\)} (B2);
+
+  \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {};
+  \draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {};
+  \node[above left, xshift=-2mm] at (Xl) {\(x\)};
+  \node[above right, xshift= 2mm] at (Xr) {\(x\)};
+\end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/lattice.tex b/buch/papers/punktgruppen/tikz/lattice.tex
new file mode 100644
index 0000000..a6b1876
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/lattice.tex
@@ -0,0 +1,40 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+\begin{tikzpicture}[
+    >=latex, 
+    dot/.style = {
+      draw, circle, thick, black, fill = gray!40!white,
+      minimum size = 2mm,
+      inner sep = 0pt,
+      outer sep = 1mm,
+    },
+  ]
+
+  \begin{scope}
+	\clip (-2,-2) rectangle (7,2);
+	\foreach \y in {-7,-6,...,7} {
+	  \foreach \x in {-7,-6,...,7} {
+		\node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {};
+	  }
+	}
+  \end{scope}
+  \draw[black, thick] (-2, -2) rectangle (7,2);
+
+  \draw[red!80!black, thick, ->] 
+	(N00) to node[midway, below] {\(\vec{a}_1\)} (N10);
+  \draw[cyan!80!black, thick, ->]
+	(N00) to node[midway, left] {\(\vec{a}_2\)} (N01);
+\end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/piezo-atoms.tex b/buch/papers/punktgruppen/tikz/piezo-atoms.tex
new file mode 100644
index 0000000..1811392
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/piezo-atoms.tex
@@ -0,0 +1,122 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex, 
+      node distance = 2mm,
+      charge/.style = {
+        circle, draw = black, thick,
+        minimum size = 5mm
+      },
+      positive/.style = { fill = red!50 },
+      negative/.style = { fill = blue!50 },
+    ]
+
+    \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum};
+
+    \begin{scope}
+      \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] {
+        \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\
+        \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+        \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\
+      };
+      \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+    \end{scope}
+
+    \begin{scope}[xshift=11cm]
+      \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} {
+        \node[charge, \t] (C\i) at (\x:1.5cm) {};
+      }
+
+      \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+      \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {};
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+    \end{scope}
+
+	%% 
+    \node[below = of title] {Polarisation Feld \(\vec{E}_p\)};
+
+	%% hex with vertical pressure
+    \begin{scope}[xshift=11cm, yshift=-4.5cm]
+      \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
+      \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
+      \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
+      \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
+      \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
+      \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
+
+      \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+      \foreach \d in {C1, C2} {
+        \draw[orange, very thick, <-] (\d) to ++(0,.7);
+      }
+
+      \foreach \d in {C4, C5} {
+        \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+      }
+
+      \node[black] (E) {\(\vec{E}_p\)};
+      \begin{scope}[node distance = .5mm]
+        \node[red!50, right = of E] {\(+\)};
+        \node[blue!50, left = of E] {\(-\)};
+      \end{scope}
+      % \draw[gray, thick, dotted] (E) to ++(0,2);
+      % \draw[gray, thick, dotted] (E) to ++(0,-2);
+    \end{scope}
+
+	%% square with vertical pressure 
+    \begin{scope}[yshift=-4.5cm]
+      \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] {
+        \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\
+        \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\
+        \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\
+      };
+
+      \foreach \d in {NW, N, NE} {
+        \draw[orange, very thick, <-] (\d) to ++(0,.7);
+      }
+
+      \foreach \d in {SW, S, SE} {
+        \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+      }
+
+      \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
+    \end{scope}
+
+	%% hex with horizontal pressure
+    \begin{scope}[xshift=5.5cm, yshift=-4.5cm]
+      \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
+      \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
+      \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
+      \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
+      \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
+      \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
+
+      \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+
+      \draw[orange, very thick, <-] (C6) to ++(.7,0);
+      \draw[orange, very thick, <-] (C3) to ++(-.7,0);
+
+      \node[black] (E) {\(\vec{E}_p\)};
+      \begin{scope}[node distance = .5mm]
+        \node[blue!50, right = of E] {\(-\)};
+        \node[red!50, left = of E] {\(+\)};
+      \end{scope}
+      % \draw[gray, thick, dotted] (E) to ++(0,2);
+      % \draw[gray, thick, dotted] (E) to ++(0,-2);
+    \end{scope}
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/piezo.tex b/buch/papers/punktgruppen/tikz/piezo.tex
new file mode 100644
index 0000000..6542f26
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/piezo.tex
@@ -0,0 +1,73 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+\begin{tikzpicture}[
+    >=latex, 
+  ]
+  \begin{scope}[
+        node distance = 0cm
+    ]
+    \node[
+        rectangle, fill = gray!20!white,
+        minimum width = 3cm, minimum height = 2cm,
+    ] (body) {\(\vec{E}_p = \vec{0}\)};
+
+    \node[
+        draw, rectangle, thick, black, fill = red!50,
+        minimum width = 3cm, minimum height = 1mm,
+        above = of body
+    ] (pos) {};
+
+    \node[
+        draw, rectangle, thick, black, fill = blue!50,
+        minimum width = 3cm, minimum height = 1mm,
+        below = of body
+    ] (neg) {};
+
+    \draw[black, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+    \draw[black, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+    \draw[black, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n);
+  \end{scope}
+  \begin{scope}[
+        node distance = 0cm,
+        xshift = 7cm
+    ]
+    \node[
+        rectangle, fill = gray!20!white,
+        minimum width = 3cm, minimum height = 1.5cm,
+    ] (body) {\(\vec{E}_p \neq \vec{0}\)};
+
+    \node[
+        draw, rectangle, thick, black, fill = red!50,
+        minimum width = 3cm, minimum height = 1mm,
+        above = of body
+    ] (pos) {};
+
+    \node[
+        draw, rectangle, thick, black, fill = blue!50,
+        minimum width = 3cm, minimum height = 1mm,
+        below = of body
+    ] (neg) {};
+
+    \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1);
+    \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1);
+
+    \draw[black, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
+    \draw[black, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
+
+    \draw[black, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n);
+  \end{scope}
+\end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/projections.tex b/buch/papers/punktgruppen/tikz/projections.tex
new file mode 100644
index 0000000..e8a4a2e
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/projections.tex
@@ -0,0 +1,258 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+\begin{tikzpicture}[
+    >=latex, 
+    classcirc/.style = {
+      draw = gray, thick, circle,
+      minimum size = 12mm,
+      inner sep = 0pt, outer sep = 0pt,
+    },
+    classlabel/.style = {
+      below right = 5mm
+    },
+    round/.style = {
+      draw = orange, thick, circle,
+      minimum size = 1mm,
+      inner sep = 0pt, outer sep = 0pt,
+    },
+    cross/.style = {
+      cross out, draw = magenta, thick,
+      minimum size = 1mm, 
+      inner sep = 0pt, outer sep = 0pt
+    },
+  ]
+  \matrix [row sep = 3mm, column sep = 0mm] {
+    \node[classcirc] (C1)  {} node[classlabel] {\(C_{1}\)};  & 
+    \node[classcirc] (C2)  {} node[classlabel] {\(C_{2}\)};  &
+    \node[classcirc] (C3)  {} node[classlabel] {\(C_{3}\)};  &
+    \node[classcirc] (Ci)  {} node[classlabel] {\(C_{i}\)};  &
+
+    \node[classcirc] (Cs)  {} node[classlabel] {\(C_{s}\)};  &
+    \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; &
+    \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; &
+    \node[classcirc] (D2)  {} node[classlabel] {\(D_{2}\)};  \\
+
+    \node[classcirc] (D3d) {} node[classlabel] {\(C_{3v}\)}; &
+    \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; & 
+    \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; & 
+    \node[classcirc] (D3)  {} node[classlabel] {\(D_{3}\)};  &
+
+    \node[classcirc] (C4)  {} node[classlabel] {\(C_{4}\)};  &
+    \node[classcirc] (C6)  {} node[classlabel] {\(C_{6}\)};  &
+    \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; &
+    \node[classcirc] (S4)  {} node[classlabel] {\(S_{4}\)};  \\
+
+    \node[classcirc] (S3)  {} node[classlabel] {\(S_{3}\)};  &
+    \node[classcirc, dashed] (T)   {} node[classlabel] {\(T_{}\)};   &
+    \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; &
+    \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; &
+
+    \node[classcirc, dashed] (Th)  {} node[classlabel] {\(T_{h}\)};  &
+    \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; &
+    \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; &
+    \node[classcirc, dashed] (Td)  {} node[classlabel] {\(T_{d}\)};  \\
+    
+    \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; &
+    \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; &
+    \node[classcirc, dashed] (O)   {} node[classlabel] {\(O_{}\)};   &
+    \node[classcirc] (D4)  {} node[classlabel] {\(D_{4}\)};  &
+
+    \node[classcirc] (D6)  {} node[classlabel] {\(D_{6}\)};  &
+    \node[classcirc, dashed] (Oh)  {} node[classlabel] {\(O_{h}\)};  &
+    \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; &
+    \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\
+  };
+
+
+  \node[cross] at ($(C1)+(4mm,0)$) {};
+
+  
+  \node[cross] at ($(C2)+(4mm,0)$) {};
+  \node[cross] at ($(C2)-(4mm,0)$) {};
+
+
+  \node[cross] at ($(C3)+(  0:4mm)$) {};
+  \node[cross] at ($(C3)+(120:4mm)$) {};
+  \node[cross] at ($(C3)+(240:4mm)$) {};
+
+
+  \node[cross] at ($(Ci)+(4mm,0)$) {};
+  \node[round] at ($(Ci)-(4mm,0)$) {};
+
+
+  \node[cross] at ($(Cs)+(4mm,0)$) {};
+  \node[round] at ($(Cs)+(4mm,0)$) {};
+
+
+  \node[cross] at ($(C3i)+(  0:4mm)$) {};
+  \node[cross] at ($(C3i)+(120:4mm)$) {};
+  \node[cross] at ($(C3i)+(240:4mm)$) {};
+  \node[round] at ($(C3i)+( 60:4mm)$) {};
+  \node[round] at ($(C3i)+(180:4mm)$) {};
+  \node[round] at ($(C3i)+(300:4mm)$) {};
+
+
+  \node[cross] at ($(C2h)+(4mm,0)$) {};
+  \node[cross] at ($(C2h)-(4mm,0)$) {};
+  \node[round] at ($(C2h)+(4mm,0)$) {};
+  \node[round] at ($(C2h)-(4mm,0)$) {};
+
+
+  \node[cross] at ($(D2)+( 20:4mm)$) {};
+  \node[cross] at ($(D2)+(200:4mm)$) {};
+  \node[round] at ($(D2)+(160:4mm)$) {};
+  \node[round] at ($(D2)+(340:4mm)$) {};
+
+
+  \foreach \x in {0, 120, 240} {
+    \node[cross] at ($(D3d)+({\x+15}:4mm)$) {};
+    \node[cross] at ($(D3d)+({\x-15}:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 180} {
+    \node[cross] at ($(C2v)+({\x+15}:4mm)$) {};
+    \node[cross] at ($(C2v)+({\x-15}:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 180} {
+    \node[cross] at ($(D2h)+({\x+15}:4mm)$) {};
+    \node[cross] at ($(D2h)+({\x-15}:4mm)$) {};
+    \node[round] at ($(D2h)+({\x+15}:4mm)$) {};
+    \node[round] at ($(D2h)+({\x-15}:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 120, 240} {
+    \node[cross] at ($(D3)+({\x+15}:4mm)$) {};
+    \node[round] at ($(D3)+({\x-15}:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 90, 180, 270} {
+    \node[cross] at ($(C4)+(\x:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 60, 120, 180, 240, 300} {
+    \node[cross] at ($(C6)+(\x:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 120, 240} {
+    \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {};
+    \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {};
+    \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {};
+    \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {};
+  }
+
+
+  \node[cross] at ($(S4)+(4mm,0)$) {};
+  \node[cross] at ($(S4)-(4mm,0)$) {};
+  \node[round] at ($(S4)+(0,4mm)$) {};
+  \node[round] at ($(S4)-(0,4mm)$) {};
+
+
+  \foreach \x in {0, 120, 240} {
+    \node[cross] at ($(S3)+(\x:4mm)$) {};
+    \node[round] at ($(S3)+(\x:4mm)$) {};
+  }
+
+
+  %% TODO: T
+
+
+  \foreach \x in {0, 90, 180, 270} {
+    \node[cross] at ($(C4h)+(\x:4mm)$) {};
+    \node[round] at ($(C4h)+(\x:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 60, 120, 180, 240, 300} {
+    \node[cross] at ($(C6h)+(\x:4mm)$) {};
+    \node[round] at ($(C6h)+(\x:4mm)$) {};
+  }
+
+
+  %% TODO: Th
+
+
+  \foreach \x in {0, 90, 180, 270} {
+    \node[cross] at ($(C4v)+(\x+15:4mm)$) {};
+    \node[cross] at ($(C4v)+(\x-15:4mm)$) {};
+  }
+
+
+
+  \foreach \x in {0, 60, 120, 180, 240, 300} {
+    \node[cross] at ($(C6v)+(\x+10:4mm)$) {};
+    \node[cross] at ($(C6v)+(\x-10:4mm)$) {};
+  }
+
+
+  %% TODO: Td
+
+
+  \foreach \x in {0, 180} {
+    \node[cross] at ($(D2d)+({\x+15}:4mm)$) {};
+    \node[round] at ($(D2d)+({\x-15}:4mm)$) {};
+
+    \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {};
+    \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 120, 240} {
+    \node[cross] at ($(D3h)+({\x+15}:4mm)$) {};
+    \node[cross] at ($(D3h)+({\x-15}:4mm)$) {};
+    \node[round] at ($(D3h)+({\x+15}:4mm)$) {};
+    \node[round] at ($(D3h)+({\x-15}:4mm)$) {};
+  }
+
+
+  %% TODO: O
+
+
+  \foreach \x in {0, 90, 180, 270} {
+    \node[cross] at ($(D4)+({\x+15}:4mm)$) {};
+    \node[round] at ($(D4)+({\x-15}:4mm)$) {};
+  }
+
+  \foreach \x in {0, 60, 120, 180, 240, 300} {
+    \node[cross] at ($(D6)+({\x+10}:4mm)$) {};
+    \node[round] at ($(D6)+({\x-10}:4mm)$) {};
+  }
+
+
+  % TODO Oh
+
+
+  \foreach \x in {0, 90, 180, 270} {
+    \node[cross] at ($(D4h)+(\x+15:4mm)$) {};
+    \node[cross] at ($(D4h)+(\x-15:4mm)$) {};
+    \node[round] at ($(D4h)+(\x+15:4mm)$) {};
+    \node[round] at ($(D4h)+(\x-15:4mm)$) {};
+  }
+
+
+  \foreach \x in {0, 60, 120, 180, 240, 300} {
+    \node[cross] at ($(D6h)+({\x+10}:4mm)$) {};
+    \node[cross] at ($(D6h)+({\x-10}:4mm)$) {};
+    \node[round] at ($(D6h)+({\x+10}:4mm)$) {};
+    \node[round] at ($(D6h)+({\x-10}:4mm)$) {};
+  }
+\end{tikzpicture}
+\end{document}
diff --git a/buch/papers/punktgruppen/tikz/stereographic-projections.tex b/buch/papers/punktgruppen/tikz/stereographic-projections.tex
new file mode 100644
index 0000000..7d612fb
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/stereographic-projections.tex
@@ -0,0 +1,108 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{tikz-3dplot}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+
+\tdplotsetmaincoords{60}{130}
+\pgfmathsetmacro{\l}{2}
+
+\begin{tikzpicture}[
+    >=latex,
+    tdplot_main_coords,
+    dot/.style = {
+      black, fill = black, circle,
+      outer sep = 0, inner sep = 0,
+      minimum size = 1mm 
+    },
+    round/.style = {
+      draw = orange, thick, circle,
+      minimum size = 1mm,
+      inner sep = 0pt, outer sep = 0pt,
+    },
+    cross/.style = {
+      cross out, draw = magenta, thick,
+      minimum size = 1mm, 
+      inner sep = 0pt, outer sep = 0pt
+    },
+  ]
+
+  % origin
+  \coordinate (O) at (0,0,0);
+
+  % poles
+  \coordinate (NP) at (0,0,\l);
+  \coordinate (SP) at (0,0,-\l);
+
+  % axis
+  % \draw[->] (O) -- ++(1.5*\l,0,0);
+  % \draw[->] (O) -- ++(0,1.5*\l,0);
+  % \draw[->] (O) -- ++(0,0,1.5*\l);
+
+  % gray unit circle
+  \tdplotdrawarc[gray, thick]{(O)}{\l}{0}{360}{}{};
+  \draw[gray, dotted] (-\l, 0, 0) to (\l, 0, 0);
+  \draw[gray, dotted] (0, -\l, 0) to (0, \l, 0);
+
+  % meridians
+  \foreach \phi in {0, 30, 60, ..., 150}{
+    \tdplotsetrotatedcoords{\phi}{90}{0};
+    \tdplotdrawarc[lightgray, dashed, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
+  }
+
+  % dot above and its projection
+  \pgfmathsetmacro{\phi}{120}
+  \pgfmathsetmacro{\theta}{60}
+
+  \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l}
+  \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l}
+  \pgfmathsetmacro{\pz}{cos(\theta)*\l})
+
+  \coordinate (A) at (\px,\py,\pz);
+  \coordinate (Aproj) at ({\px * \l / (\l + \pz)}, {\py * \l / (\l + \pz)}, 0);
+
+  % dot below and its projection
+  \pgfmathsetmacro{\phi}{-60}
+  \pgfmathsetmacro{\theta}{120}
+
+  \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l}
+  \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l}
+  \pgfmathsetmacro{\pz}{cos(\theta)*\l})
+
+  \coordinate (B) at (\px,\py,\pz);
+  \coordinate (Bproj) at ({\px * \l / (\l - \pz)}, {\py * \l / (\l - \pz)}, 0);
+
+  % projection lines
+  \draw[gray] (A) to (SP);
+  \draw[gray] (SP) to (O) to (Aproj);
+
+  \draw[gray] (B) to (NP);
+  \draw[gray] (NP) to (O) to (Bproj);
+
+  % dots
+  \draw (O) node[dot] {};
+  \draw (SP) node[dot] {};
+  \draw (NP) node[dot] {};
+  \draw (A) node[dot, fill = magenta, minimum size = 1.5mm] {};
+  \draw (B) node[dot, fill = orange, minimum size = 1.5mm] {};
+
+  % projection markers
+  \draw[very thick, magenta] 
+    (Aproj) ++(.15,0) to ($(Aproj)+(-.15, 0)$)
+    (Aproj) ++(0,.15) to ($(Aproj) +(0, -.15)$);
+
+  \tdplotdrawarc[orange, very thick]{(Bproj)}{.1}{0}{360}{}{};
+
+\end{tikzpicture}
+\end{document}
+% vim:ts=2 sw=2 et:
diff --git a/buch/papers/punktgruppen/tikz/symmetric-shapes.tex b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex
new file mode 100644
index 0000000..688fb61
--- /dev/null
+++ b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex
@@ -0,0 +1,60 @@
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+
+\usetikzlibrary{arrows}
+\usetikzlibrary{shapes.geometric}
+\usetikzlibrary{intersections}
+\usetikzlibrary{math}
+\usetikzlibrary{positioning}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{shapes.misc}
+\usetikzlibrary{calc}
+
+\begin{document}
+	\begin{tikzpicture}[
+      >=latex, 
+			node distance = 2cm,
+			shapetheme/.style = {
+				very thick, draw = black, fill = magenta!20!white,
+				minimum size = 2cm,
+			},
+			line/.style = {thick, draw = darkgray},
+			axis/.style = {line, dashed},
+			dot/.style = {
+				circle, draw = darkgray, fill = darkgray,
+				minimum size = 1mm, inner sep = 0, outer sep = 0,
+			},
+		]
+
+		\node[
+			shapetheme, rectangle
+		] (R) {};
+		\node[dot] at (R) {};
+		\draw[axis] (R) ++(-1.5, 0) to ++(3, 0) node[right] {\(\sigma\)};
+
+		\node[
+			shapetheme,
+			regular polygon,
+			regular polygon sides = 5,
+			right = of R,
+		] (Ps) {};
+		\node[dot] (P) at (Ps) {};
+		\draw[line, dotted] (P) to ++(18:1.5);
+		\draw[line, dotted] (P) to ++(90:1.5);
+		\draw[line, ->] (P) ++(18:1.2) 
+			arc (18:90:1.2) node[midway, above right] {\(r, 72^\circ\)};
+
+		\node[
+			shapetheme,
+			circle, right = of P
+		] (Cs) {};
+		\node[dot] (C) at (Cs) {};
+		\draw[line, dotted] (C) to ++(1.5,0);
+		\draw[line, dotted] (C) to ++(60:1.5);
+		\draw[line, ->] (C) ++(1.2,0)
+			arc (0:60:1.2) node[midway, above right] {\(r, \alpha\)};
+
+	\end{tikzpicture}
+\end{document}