Merge branch 'master' into master

14 files changed, 911 insertions, 0 deletions

diff --git a/buch/chapters/60-gruppen/images/Makefile b/buch/chapters/60-gruppen/images/Makefile
new file mode 100644
index 0000000..3ed39e5
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/Makefile
@@ -0,0 +1,25 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	phasenraum.pdf kartenkreis.pdf karten.pdf sl2.pdf scherungen.pdf
+
+phasenraum.pdf:	phasenraum.tex
+	pdflatex phasenraum.tex
+
+kartenkreis.pdf:	kartenkreis.tex
+	pdflatex kartenkreis.tex
+
+torus.png:	torus.pov
+	povray +A0.1 -W1920 -H1080 -Otorus.png torus.pov
+
+karten.pdf:	karten.tex torus.png
+	pdflatex karten.tex
+
+sl2.pdf:	sl2.tex 
+	pdflatex sl2.tex
+
+scherungen.pdf:	scherungen.tex 
+	pdflatex scherungen.tex
+
diff --git a/buch/chapters/60-gruppen/images/castle.jpeg b/buch/chapters/60-gruppen/images/castle.jpeg
new file mode 100644
index 0000000..bf90a36
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/castle.jpeg
diff --git a/buch/chapters/60-gruppen/images/karten.pdf b/buch/chapters/60-gruppen/images/karten.pdf
new file mode 100644
index 0000000..d769cca
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/karten.pdf
diff --git a/buch/chapters/60-gruppen/images/karten.tex b/buch/chapters/60-gruppen/images/karten.tex
new file mode 100644
index 0000000..c8eb4a3
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/karten.tex
@@ -0,0 +1,112 @@
+%
+% karten.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\node at (0,0) {\includegraphics[width=10cm]{torus.png}};
+
+\def\s{3}
+
+\node at (-3.5,-0.4) {$U_\alpha$};
+\node at (2.0,-0.4) {$U_\beta$};
+
+\draw[->] (-2,-2.2) -- (-3,-4.3);
+\node at (-2.5,-3.25) [left] {$\varphi_\alpha$};
+
+\draw[->] (1.4,-1.7) -- (3,-4.3);
+\node at (2.5,-3.25) [right] {$\varphi_\beta$};
+
+\begin{scope}[xshift=-4.5cm,yshift=-8cm]
+	\begin{scope}
+		\clip (0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
+		\begin{scope}[xshift=1.8cm,yshift=0.6cm,rotate=30]
+			\fill[color=gray!20]
+				(0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
+			\foreach \x in {0,0.2,...,1}{
+				\draw[color=darkgreen]
+					({\x*\s},{-0.2*\s})
+					--
+					({\x*\s},{1.2*\s});
+			}
+			\foreach \y in {-0.2,0,...,1.2}{
+				\draw[color=orange]
+					(0,{\y*\s})
+					--
+					({1*\s},{\y*\s});
+			}
+		\end{scope}
+	\end{scope}
+
+	\foreach \x in {0,0.2,...,1}{
+		\draw[color=blue,line width=1.4pt]
+			({\x*\s},{-0.2*\s}) -- ({\x*\s},{1.2*\s});
+	}
+	\foreach \y in {-0.2,0,...,1.2}{
+		\draw[color=red,line width=1.4pt]
+			(0,{\y*\s}) -- ({1*\s},{\y*\s});
+	}
+
+	\draw[->] ({\s*(-0.1)},0) -- ({1.1*\s},0) coordinate[label={$x_1$}];
+	\draw[->] (0,{-0.3*\s}) -- (0,{1.3*\s}) coordinate[label={left:$x_2$}];
+
+	\node at ({1*\s},{1.2*\s}) [above right] {$\mathbb{R}^2$};
+
+\end{scope}
+
+\begin{scope}[xshift=1.5cm,yshift=-8cm]
+	\begin{scope}
+		\clip (0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
+		% x = - [ (sqrt(3)/2)*0.6+(1/2)*0.2 ] = -0.6196
+		% y = - [ (-1/2)*0.6 + (sqrt(3)/2)*0.2 ] = 
+		\begin{scope}[xshift=-1.8588cm,yshift=0.3804cm,rotate=-30]
+			\fill[color=gray!20]
+				(0,{-0.2*\s}) rectangle ({1*\s},{1.2*\s});
+				\foreach \x in {0,0.2,...,1}{
+					\draw[color=blue]
+						({\x*\s},{-0.2*\s})
+						--
+						({\x*\s},{1.2*\s});
+				}
+				\foreach \y in {-0.2,0,...,1.2}{
+					\draw[color=red]
+						(0,{\y*\s})
+						--
+						({1*\s},{\y*\s});
+				}
+		\end{scope}
+	\end{scope}
+
+	\foreach \x in {0,0.2,...,1}{
+		\draw[color=darkgreen,line width=1.4pt]
+			({\x*\s},{-0.2*\s}) -- ({\x*\s},{1.2*\s});
+	}
+	\foreach \y in {-0.2,0,...,1.2}{
+		\draw[color=orange,line width=1.4pt] (0,{\y*\s}) -- ({1*\s},{\y*\s});
+	}
+	\draw[->] ({\s*(-0.1)},0) -- ({1.1*\s},0) coordinate[label={$x_1$}];
+	\draw[->] (0,{-0.3*\s}) -- (0,{1.3*\s}) coordinate[label={left:$x_2$}];
+	\node at ({1*\s},{1.2*\s}) [above right] {$\mathbb{R}^2$};
+\end{scope}
+
+\draw[<-,color=white,opacity=0.8,line width=5pt] (2.5,-6.5) arc (55:100:6.5);
+\draw[<-,shorten >= 0.1cm,shorten <= 0.3cm] (2.5,-6.5) arc (55:100:6.5);
+
+\node at (0,-5.9)
+	{$\varphi_{\beta\alpha}=\varphi_\beta\circ\varphi_\alpha^{-1}$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/60-gruppen/images/kartenkreis.pdf b/buch/chapters/60-gruppen/images/kartenkreis.pdf
new file mode 100644
index 0000000..4619b56
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/kartenkreis.pdf
diff --git a/buch/chapters/60-gruppen/images/kartenkreis.tex b/buch/chapters/60-gruppen/images/kartenkreis.tex
new file mode 100644
index 0000000..4f19937
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/kartenkreis.tex
@@ -0,0 +1,189 @@
+%
+% kartenkreis.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{3}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\fill[color=red!20] (0,-1) rectangle (1.5,1);
+\fill[color=blue!20] (-1.5,-1) rectangle (0,1);
+\fill[color=darkgreen!40,opacity=0.5] (-1,0) rectangle (1,1.5);
+\fill[color=orange!40,opacity=0.5] (-1,-1.5) rectangle (1,0);
+\fill[color=white] (0,0) circle[radius=1];
+
+\fill[color=gray!20]
+	(0,-1.5) -- (0.02,-1.6) -- (0.5,-1.8) -- (0.98,-1.6) -- (1,-1.5)
+	-- cycle;
+\fill[color=gray!20]
+	(0,1.5) -- (0.02,1.6) -- (0.5,1.8) -- (0.98,1.6) -- (1,1.5)
+	-- cycle;
+\fill[color=gray!20]
+	(0,-1.5) -- (-0.02,-1.6) -- (-0.5,-1.8) -- (-0.98,-1.6) -- (-1,-1.5)
+	-- cycle;
+\fill[color=gray!20]
+	(0,1.5) -- (-0.02,1.6) -- (-0.5,1.8) -- (-0.98,1.6) -- (-1,1.5)
+	-- cycle;
+
+\fill[color=gray!20]
+	(1.5,0) -- (1.6,0.02) -- (1.8,0.5) -- (1.6,0.98) -- (1.5,1)
+	-- cycle;
+\fill[color=gray!20]
+	(-1.5,0) -- (-1.6,0.02) -- (-1.8,0.5) -- (-1.6,0.98) -- (-1.5,1)
+	-- cycle;
+\fill[color=gray!20]
+	(1.5,0) -- (1.6,-0.02) -- (1.8,-0.5) -- (1.6,-0.98) -- (1.5,-1)
+	-- cycle;
+\fill[color=gray!20]
+	(-1.5,0) -- (-1.6,-0.02) -- (-1.8,-0.5) -- (-1.6,-0.98) -- (-1.5,-1)
+	-- cycle;
+
+\draw[->] (0.5,-1.8) arc (-180:-90:0.1) arc (-90:0:1.3) arc (0:90:0.1);
+\draw[->] (1.8,0.5) arc (-90:0:0.1) arc (0:90:1.3) arc (90:180:0.1);
+\draw[->] (-0.5,1.8) arc (0:90:0.1) arc (90:180:1.3) arc (180:270:0.1);
+\draw[->] (-1.8,-0.5) arc (90:180:0.1) arc (180:270:1.3) arc (270:360:0.1);
+
+\node at (1.01,1.32)
+	[right] {$\varphi_3\circ \varphi_1^{-1}(y)=\sqrt{1-y^2}$};
+\node at (1.6,1.6) {$\varphi_{31}$};
+
+\node at (1.01,-1.28)
+	[right] {$\varphi_1\circ \varphi_4^{-1}(x)=-\sqrt{1-x^2}$};
+\node at (1.6,-1.6) {$\varphi_{14}$};
+
+\node at (-1.24,1.32)
+	[left] {$\varphi_2\circ\varphi_3^{-1}(x)=\sqrt{1-x^2}$};
+\node at (-1.6,1.6) {$\varphi_{23}$};
+
+\node at (-1.18,-1.28)
+	[left] {$\varphi_4\circ\varphi_2^{-1}(y)=-\sqrt{1-y^2}$};
+\node at (-1.6,-1.6) {$\varphi_{42}$};
+
+
+\foreach \y in {0.1,0.3,...,0.9}{
+	\draw[->,color=red,shorten >= 0.1cm,shorten <= 0.3cm]
+		({sqrt(1-\y*\y)},{\y}) -- (1.5,\y);
+	\draw[->,color=red,shorten >= 0.1cm,shorten <= 0.3cm]
+		({sqrt(1-\y*\y)},{-\y}) -- (1.5,-\y);
+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.3cm]
+		({-sqrt(1-\y*\y)},{\y}) -- (-1.5,\y);
+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.3cm]
+		({-sqrt(1-\y*\y)},{-\y}) -- (-1.5,-\y);
+}
+\foreach \x in {0.1,0.3,...,0.9}{
+	\draw[->,color=darkgreen,shorten >= 0.1cm,shorten <= 0.3cm]
+		({\x},{sqrt(1-\x*\x)}) -- ({\x},1.5);
+	\draw[->,color=darkgreen,shorten >= 0.1cm,shorten <= 0.3cm]
+		({-\x},{sqrt(1-\x*\x)}) -- ({-\x},1.5);
+	\draw[->,color=orange,shorten >= 0.1cm,shorten <= 0.3cm]
+		({\x},{-sqrt(1-\x*\x)}) -- ({\x},-1.5);
+	\draw[->,color=orange,shorten >= 0.1cm,shorten <= 0.3cm]
+		({-\x},{-sqrt(1-\x*\x)}) -- ({-\x},-1.5);
+}
+
+%\draw[color=gray!50,line width=3pt] (0,0) circle[radius=1];
+\draw[color=yellow!30,line width=3pt] (0,0) circle[radius=1];
+\node[color=yellow] at ({1/sqrt(2)},{1/sqrt(2)}) [above right] {$S^1$};
+
+\def\r{1.02}
+
+\begin{scope}
+	\clip (0,-1.1) rectangle (1.1,1.1);
+	\draw[color=red,line width=1.4pt] (-89:\r) arc (-89:89:\r);
+	\draw[color=red,line width=1.4pt] (0,-\r) circle[radius=0.02];
+	\draw[color=red,line width=1.4pt] (0,\r) circle[radius=0.02];
+\end{scope}
+
+\begin{scope}
+	\clip (-1.1,-1.1) rectangle (0,1.1);
+	\draw[color=blue,line width=1.4pt] (91:\r) arc (91:269:\r);
+	\draw[color=blue,line width=1.4pt] (0,-\r) circle[radius=0.02];
+	\draw[color=blue,line width=1.4pt] (0,\r) circle[radius=0.02];
+\end{scope}
+
+\xdef\r{0.98}
+
+\begin{scope} 
+	\clip (-1.1,0) rectangle (1.1,1.1);
+	\draw[color=darkgreen,line width=1.4pt] (1:\r) arc (1:179:\r);
+	\draw[color=darkgreen,line width=1.4pt] (\r,0) circle[radius=0.02];
+	\draw[color=darkgreen,line width=1.4pt] (-\r,0) circle[radius=0.02];
+\end{scope}
+
+\begin{scope} 
+	\clip (-1.1,-1.1) rectangle (1.1,0);
+	\draw[color=orange,line width=1.4pt] (181:\r) arc (181:359:\r);
+	\draw[color=orange,line width=1.4pt] (\r,0) circle[radius=0.02];
+	\draw[color=orange,line width=1.4pt] (-\r,0) circle[radius=0.02];
+\end{scope}
+
+\begin{scope}[yshift=1.5cm]
+	\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={$\mathbb{R}$}];
+	\begin{scope}
+		\clip (-1,-0.1) rectangle (1,0.1);
+		\draw[color=darkgreen,line width=1.4pt] (-0.98,0) -- (0.98,0);
+		\draw[color=darkgreen,line width=1.4pt] (-1,0)
+			circle[radius=0.02];
+		\draw[color=darkgreen,line width=1.4pt] (1,0)
+			circle[radius=0.02];
+	\end{scope}
+\end{scope}
+
+\begin{scope}[yshift=-1.5cm]
+	\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={below:$\mathbb{R}$}];
+	\begin{scope}
+		\clip (-1,-0.1) rectangle (1,0.1);
+		\draw[color=orange,line width=1.4pt] (-0.98,0) -- (0.98,0);
+		\draw[color=orange,line width=1.4pt] (-1,0) circle[radius=0.02];
+		\draw[color=orange,line width=1.4pt] (1,0) circle[radius=0.02];
+	\end{scope}
+\end{scope}
+
+\begin{scope}[xshift=1.5cm]
+	\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={right:$\mathbb{R}$}];
+	\begin{scope}
+		\clip (-0.1,-1) rectangle (0.1,1);
+		\draw[color=red,line width=1.4pt] (0,-0.98) -- (0,0.98);
+		\draw[color=red,line width=1.4pt] (0,-1) circle[radius=0.02];
+		\draw[color=red,line width=1.4pt] (0,1) circle[radius=0.02];
+	\end{scope}
+\end{scope}
+
+\begin{scope}[xshift=-1.5cm]
+	\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={left:$\mathbb{R}$}];
+	\begin{scope}
+		\clip (-0.1,-1) rectangle (0.1,1);
+		\draw[color=blue,line width=1.4pt] (0,-0.98) -- (0,0.98);
+		\draw[color=blue,line width=1.4pt] (0,-1) circle[radius=0.02];
+		\draw[color=blue,line width=1.4pt] (0,1) circle[radius=0.02];
+	\end{scope}
+\end{scope}
+
+\node[color=red] at (23:1) [right] {$U_{x>0}$};
+\node[color=red] at (1.25,0) [right] {$\varphi_1$};
+
+\node[color=blue] at (157:1) [left] {$U_{x<0}$};
+\node[color=blue] at (-1.25,0) [left] {$\varphi_2$};
+
+\node[color=darkgreen] at (115:1) [below right] {$U_{y>0}$};
+\node[color=darkgreen] at (0,1.25) [above] {$\varphi_3$};
+
+\node[color=orange] at (-115:1) [above right] {$U_{y<0}$};
+\node[color=orange] at (0,-1.25) [below] {$\varphi_4$};
+
+\draw[->] (-1.1,0) -- (1.15,0) coordinate[label={$x$}];
+\draw[->] (0,-1.1) -- (0,1.15) coordinate[label={right:$y$}];
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/60-gruppen/images/phasenraum.pdf b/buch/chapters/60-gruppen/images/phasenraum.pdf
new file mode 100644
index 0000000..adfb0c6
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/phasenraum.pdf
diff --git a/buch/chapters/60-gruppen/images/phasenraum.tex b/buch/chapters/60-gruppen/images/phasenraum.tex
new file mode 100644
index 0000000..2bccc27
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/phasenraum.tex
@@ -0,0 +1,93 @@
+%
+% phasenraum.tex -- 
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\m{1}
+\def\K{0.444}
+
+\pgfmathparse{sqrt(\K/\m)}
+\xdef\o{\pgfmathresult}
+
+\def\punkt#1#2{ ({#2*cos(#1)},{\o*#2*sin(#1)}) }
+
+\foreach \r in {0.5,1,...,6}{
+	\draw plot[domain=0:359,samples=360]
+			({\r*cos(\x)},{\o*\r*sin(\x)}) -- cycle;
+}
+
+\def\tangente#1#2{
+		\pgfmathparse{#2/\m}
+		\xdef\u{\pgfmathresult}
+
+		\pgfmathparse{-#1*\K}
+		\xdef\v{\pgfmathresult}
+
+		\pgfmathparse{sqrt(\u*\u+\v*\v)}
+		\xdef\l{\pgfmathresult}
+
+		\fill[color=blue] (#1,#2) circle[radius=0.03];
+		\draw[color=blue,line width=0.5pt]
+			({#1-0.2*\u/\l},{#2-0.2*\v/\l})
+			--
+			({#1+0.2*\u/\l},{#2+0.2*\v/\l});
+}
+
+\foreach \x in {-6.25,-5.75,...,6.3}{
+	\foreach \y in {-4.25,-3.75,...,4.3}{
+		\tangente{\x}{\y}
+	}
+}
+
+%\foreach \x in {0.5,1,...,5.5,6}{
+%	\tangente{\x}{0}
+%	\tangente{-\x}{0}
+%	\foreach \y in {0.5,1,...,4}{
+%		\tangente{\x}{\y}
+%		\tangente{-\x}{\y}
+%		\tangente{\x}{-\y}
+%		\tangente{-\x}{-\y}
+%	}
+%}
+%\foreach \y in {0.5,1,...,4}{
+%	\tangente{0}{\y}
+%	\tangente{0}{-\y}
+%}
+
+\fill[color=white,opacity=0.7] \punkt{60}{4} rectangle \punkt{59}{5.8};
+\fill[color=white,opacity=0.7] \punkt{0}{4} rectangle \punkt{18}{4.9};
+
+\draw[->,color=red,line width=1.4pt]
+	plot[domain=0:60,samples=360]
+		({4*cos(\x)},{\o*4*sin(\x)});
+
+\draw[->] (-6.5,0) -- (6.7,0) coordinate[label={$x$}];
+\draw[->] (0,-4.5) -- (0,4.7) coordinate[label={right:$p$}];
+
+\fill[color=red] \punkt{60}{4} circle[radius=0.08];
+\node[color=red] at \punkt{60}{4} [above right]
+	{$\begin{pmatrix}x(t)\\p(t)\end{pmatrix}$};
+
+\fill[color=red] \punkt{0}{4} circle[radius=0.08];
+\node[color=red] at \punkt{0}{4} [above right]
+	{$\begin{pmatrix}x_0\\0\end{pmatrix}$};
+
+\fill[color=white] (4,0) circle[radius=0.05];
+\node at (3.9,0) [below right] {$x_0$};
+\fill (0,{\o*4}) circle[radius=0.05];
+\node at (0.1,{\o*4+0.05}) [below left] {$\omega x_0$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/60-gruppen/images/scherungen.pdf b/buch/chapters/60-gruppen/images/scherungen.pdf
new file mode 100644
index 0000000..c7f4988
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/scherungen.pdf
diff --git a/buch/chapters/60-gruppen/images/scherungen.tex b/buch/chapters/60-gruppen/images/scherungen.tex
new file mode 100644
index 0000000..893bd12
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/scherungen.tex
@@ -0,0 +1,157 @@
+%
+% scherungen.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{blau}{rgb}{0,0.8,1}
+\definecolor{blau}{rgb}{0,0.6,0}
+\def\s{1.1}
+
+\begin{scope}[xshift=-4.6cm]
+
+	\fill[color=blue!20] (0,0) rectangle (2,2);
+	\fill[color=red!40,opacity=0.5] (0,0) -- (2,\s) -- (2,{2+\s}) -- (0,2)
+		-- cycle;
+
+	\foreach \x in {-1,...,3}{
+		\draw[color=blau]  (\x,-1) -- (\x,3);
+		\draw[color=blau]  (-1,\x) -- (3,\x);
+	}
+
+	\begin{scope}
+		\clip (-1,-1) rectangle (3,3);
+		\foreach \x in {-1,...,3}{
+			\draw[color=orange] (\x,-1) -- (\x,3);
+			\draw[color=orange] (-1,{\x-0.5*\s}) -- (3,{\x+1.5*\s});
+		}
+	\end{scope}
+
+	\draw[->] (-1.1,0) -- (3.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];
+
+	\node[color=blue] at (0,2) [above left] {$1$};
+	\node[color=blue] at (2,0) [below right] {$1$};
+	\draw[->,color=blue] (0,0) -- (2,0);
+	\draw[->,color=blue] (0,0) -- (0,2);
+
+	\draw[->,color=red] (0,0) -- (2,\s);
+	\draw[->,color=red] (0,0) -- (0,2);
+
+	\node[color=red] at (2,\s) [below right] {$(1,t)$};
+
+	\node at (0,0) [below right] {$O$};
+	\node at (1,-1.1) [below] {$\displaystyle
+		\begin{aligned}
+		M &= \begin{pmatrix}0&0\\1&0 \end{pmatrix}
+		\\
+		e^{Mt}
+		&=
+		\begin{pmatrix}1&0\\t&1 \end{pmatrix}
+		\end{aligned}
+	$};
+\end{scope}
+
+\begin{scope}
+	\fill[color=blue!20] (0,0) rectangle (2,2);
+	\fill[color=red!40,opacity=0.5] (0,0) -- (2,0) -- ({2+\s},2) -- (\s,2)
+		-- cycle;
+
+	\foreach \x in {-1,...,3}{
+		\draw[color=blau]  (\x,-1) -- (\x,3);
+		\draw[color=blau]  (-1,\x) -- (3,\x);
+	}
+
+	\begin{scope}
+		\clip (-1,-1) rectangle (3,3);
+		\foreach \x in {-1,...,3}{
+			\draw[color=orange] (-1,\x) -- (3,\x);
+			\draw[color=orange] ({\x-0.5*\s},-1) -- ({\x+1.5*\s},3);
+		}
+	\end{scope}
+
+	\draw[->] (-1.1,0) -- (3.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];
+
+	\node[color=blue] at (0,2) [above left] {$1$};
+	\node[color=blue] at (2,0) [below right] {$1$};
+	\draw[->,color=blue] (0,0) -- (2,0);
+	\draw[->,color=blue] (0,0) -- (0,2);
+
+	\draw[->,color=red] (0,0) -- (2,0);
+	\draw[->,color=red] (0,0) -- (\s,2);
+
+	\node[color=red] at (\s,2) [above left] {$(t,1)$};
+
+	\node at (0,0) [below right] {$O$};
+	
+	\node at (1,-1.1) [below] {$\displaystyle
+		\begin{aligned} N &= \begin{pmatrix}0&1\\0&0 \end{pmatrix}
+		\\
+		e^{Nt}
+		&=
+		\begin{pmatrix}1&t\\0&1 \end{pmatrix}
+		\end{aligned}
+	$};
+\end{scope}
+
+\begin{scope}[xshift=3.6cm,yshift=0cm]
+	\def\punkt#1#2{({1.6005*(#1)+0.4114*(#2)},{-0.2057*(#1)+0.5719*(#2)})}
+	\fill[color=blue!20] (0,0) rectangle (2,2);
+	\fill[color=red!40,opacity=0.5]
+		(0,0) -- \punkt{2}{0} -- \punkt{2}{2} -- \punkt{0}{2} -- cycle;
+
+	\foreach \x in {0,...,4}{
+		\draw[color=blau]  (\x,-1) -- (\x,3);
+	}
+	\foreach \y in {-1,...,3}{
+		\draw[color=blau]  (0,\y) -- (4,\y);
+	}
+
+	\begin{scope}
+		\clip (-0,-1) rectangle (4,3);
+		\foreach \x in {-1,...,6}{
+			\draw[color=orange] \punkt{\x}{-3} -- \punkt{\x}{6};
+			\draw[color=orange] \punkt{-3}{\x} -- \punkt{6}{\x};
+		}
+	\end{scope}
+
+	\draw[->] (-0.1,0) -- (4.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-1.1) -- (0,3.5) coordinate[label={right:$y$}];
+
+	\node[color=blue] at (0,2) [above left] {$1$};
+	\node[color=blue] at (2,0) [below right] {$1$};
+	\draw[->,color=blue] (0,0) -- (2,0);
+	\draw[->,color=blue] (0,0) -- (0,2);
+
+	\draw[->,color=red] (0,0) -- \punkt{2}{0};
+	\draw[->,color=red] (0,0) -- \punkt{0}{2};
+
+	\node at (0,0) [below right] {$O$};
+	
+	\node at (2,-1.1) [below] {$\displaystyle
+		\begin{aligned} D &= \begin{pmatrix}0.5&0.4\\-0.2&-0.5 \end{pmatrix}
+		\\
+		e^{D\cdot 1}
+		&=
+		\begin{pmatrix}
+			   1.6005 & 0.4114\\
+			  -0.2057 & 0.5719
+		\end{pmatrix}
+		\end{aligned}
+	$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/60-gruppen/images/sl2.pdf b/buch/chapters/60-gruppen/images/sl2.pdf
new file mode 100644
index 0000000..ffc0759
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/sl2.pdf
diff --git a/buch/chapters/60-gruppen/images/sl2.tex b/buch/chapters/60-gruppen/images/sl2.tex
new file mode 100644
index 0000000..0e44aa9
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/sl2.tex
@@ -0,0 +1,146 @@
+%
+% sl2.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\begin{scope}[xshift=-4.5cm]
+	\fill[color=blue!20]
+		(1.4,0) -- (0,1.4) -- (-1.4,0) -- (0,-1.4) -- cycle;
+	\fill[color=red!40,opacity=0.5]
+		(1.96,0) -- (0,1) -- (-1.96,0) -- (0,-1) -- cycle;
+
+	\begin{scope}
+		\clip (-2.1,-2.1) rectangle (2.3,2.3);
+		\draw[color=darkgreen]
+			plot[domain=-1:1,samples=100]
+				({(1/1.4)*exp(\x)},{(1/1.4)*exp(-\x)});
+		\draw[color=darkgreen]
+			plot[domain=-1:1,samples=100]
+				({(1/1.4)*exp(\x)},{-(1/1.4)*exp(-\x)});
+		\draw[color=darkgreen]
+			plot[domain=-1:1,samples=100]
+				({-(1/1.4)*exp(\x)},{(1/1.4)*exp(-\x)});
+		\draw[color=darkgreen]
+			plot[domain=-1:1,samples=100]
+				({-(1/1.4)*exp(\x)},{-(1/1.4)*exp(-\x)});
+	\end{scope}
+
+	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+
+	\draw[->,color=blue] (0,0) -- (1.4,0);
+	\draw[->,color=blue] (0,0) -- (0,1.4);
+
+	\draw[->,color=red] (0,0) -- (1.96,0);
+	\draw[->,color=red] (0,0) -- (0,1);
+	\node at (0,-3.2) 
+		{$\displaystyle
+		\begin{aligned}
+		A&=\begin{pmatrix}1&0\\0&-1\end{pmatrix}
+		\\
+		e^{At}
+		&=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}
+		\end{aligned}
+		$};
+	
+\end{scope}
+
+
+\begin{scope}
+	\fill[color=blue!20]
+		(0:1.4) -- (90:1.4) -- (180:1.4) -- (270:1.4) -- cycle;
+	\fill[color=red!40,opacity=0.5]
+		(33:1.4) -- (123:1.4) -- (213:1.4) -- (303:1.4) -- cycle;
+
+	\draw[color=darkgreen] (0,0) circle[radius=1.4];
+
+	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+
+	\draw[->,color=blue] (0,0) -- (1.4,0);
+	\draw[->,color=blue] (0,0) -- (0,1.4);
+
+	\draw[->,color=red] (0,0) -- (33:1.4);
+	\draw[->,color=red] (0,0) -- (123:1.4);
+
+	\node at (0,-3.2)
+		{$\displaystyle
+		\begin{aligned}
+		B
+		&=\begin{pmatrix}0&-1\\1&0 \end{pmatrix}
+		\\
+		e^{Bt}
+		&=
+		\begin{pmatrix}
+		\cos t&-\sin t\\
+		\sin t& \cos t
+		\end{pmatrix}
+		\end{aligned}$};
+\end{scope}
+
+
+\begin{scope}[xshift=4.5cm]
+	\fill[color=blue!20]
+		(0:1.4) -- (90:1.4) -- (180:1.4) -- (270:1.4) -- cycle;
+	\def\x{0.5}
+	\fill[color=red!40,opacity=0.5]
+		({1.4*cosh(\x)},{1.4*sinh(\x})
+		--
+		({1.4*sinh(\x},{1.4*cosh(\x)})
+		--
+		({-1.4*cosh(\x)},{-1.4*sinh(\x})
+		--
+		({-1.4*sinh(\x},{-1.4*cosh(\x)})
+		-- cycle;
+
+	\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.2,2.2);
+	\draw[color=darkgreen]
+		plot[domain=-1:1,samples=100] ({1.4*cosh(\x)},{1.4*sinh(\x)});
+	\draw[color=darkgreen]
+		plot[domain=-1:1,samples=100] ({1.4*sinh(\x)},{1.4*cosh(\x)});
+	\draw[color=darkgreen]
+		plot[domain=-1:1,samples=100] ({-1.4*cosh(\x)},{1.4*sinh(\x)});
+	\draw[color=darkgreen]
+		plot[domain=-1:1,samples=100] ({1.4*sinh(\x)},{-1.4*cosh(\x)});
+	\end{scope}
+
+	\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+
+	\draw[->,color=blue] (0,0) -- (1.4,0);
+	\draw[->,color=blue] (0,0) -- (0,1.4);
+
+	\draw[->,color=red] (0,0) -- ({1.4*cosh(\x)},{1.4*sinh(\x)});
+	\draw[->,color=red] (0,0) -- ({1.4*sinh(\x)},{1.4*cosh(\x)});
+
+	\node at (0,-3.2) {$\displaystyle
+		\begin{aligned}
+		C&=\begin{pmatrix}0&1\\1&0\end{pmatrix}
+		\\
+		e^{Ct}
+		&=
+		\begin{pmatrix}
+		\cosh t&\sinh t\\
+		\sinh t&\cosh t
+		\end{pmatrix}
+		\end{aligned}
+		$};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/60-gruppen/images/torus.png b/buch/chapters/60-gruppen/images/torus.png
new file mode 100644
index 0000000..c42440f
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/torus.png
diff --git a/buch/chapters/60-gruppen/images/torus.pov b/buch/chapters/60-gruppen/images/torus.pov
new file mode 100644
index 0000000..3a8e327
--- /dev/null
+++ b/buch/chapters/60-gruppen/images/torus.pov
@@ -0,0 +1,189 @@
+//
+// diffusion.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.034;
+#declare N = 100;
+#declare r = 0.43;
+#declare R = 1;
+
+camera {
+        location <43, 25, -20>
+        look_at <0, -0.01, 0>
+        right 16/9 * x * imagescale
+        up y * imagescale
+}
+
+light_source {
+        <10, 20, -40> color White
+        area_light <1,0,0> <0,0,1>, 10, 10
+        adaptive 1
+        jitter
+}
+
+sky_sphere {
+        pigment {
+                color rgb<1,1,1>
+        }
+}
+
+#macro rotiere(phi, vv)
+	< cos(phi) * vv.x - sin(phi) * vv.z, vv.y, sin(phi) * vv.x + cos(phi) * vv.z >
+#end
+
+#macro punkt(phi,theta)
+	rotiere(phi, < R + r * cos(theta), r * sin(theta), 0 >)
+#end
+
+mesh {
+	#declare phistep = 2 * pi / N;
+	#declare thetastep = 2 * 2 * pi / N;
+	#declare phi = 0;
+	#while (phi < 2 * pi - phistep/2)
+		#declare theta = 0;
+		#while (theta < 2 * pi - thetastep/2)
+			triangle {
+				punkt(phi          , theta            ), 
+				punkt(phi + phistep, theta            ), 
+				punkt(phi + phistep, theta + thetastep)
+			}
+			triangle {
+				punkt(phi          , theta            ), 
+				punkt(phi + phistep, theta + thetastep), 
+				punkt(phi          , theta + thetastep)
+			}
+			#declare theta = theta + thetastep;
+		#end
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color Gray
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+#declare thetastart = -0.2;
+#declare thetaend = 1.2;
+#declare phistart = 5;
+#declare phiend = 6;
+
+union {
+	#declare thetastep = 0.2;
+	#declare theta = thetastart;
+	#while (theta < thetaend + thetastep/2)
+		#declare phistep = (phiend-phistart)/N;
+		#declare phi = phistart;
+		#while (phi < phiend - phistep/2)
+			sphere { punkt(phi,theta), 0.01 }
+			cylinder {
+				punkt(phi,theta),
+				punkt(phi+phistep,theta),
+				0.01
+			}
+			#declare phi = phi + phistep;
+		#end
+		sphere { punkt(phi,theta), 0.01 }
+		#declare theta = theta + thetastep;
+	#end
+
+	pigment {
+		color Red
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+union {
+	#declare phistep = 0.2;
+	#declare phi = phistart;
+	#while (phi < phiend + phistep/2)
+		#declare thetastep = (thetaend-thetastart)/N;
+		#declare theta = thetastart;
+		#while (theta < thetaend - thetastep/2)
+			sphere { punkt(phi,theta), 0.01 }
+			cylinder {
+				punkt(phi,theta),
+				punkt(phi,theta+thetastep),
+				0.01
+			}
+			#declare theta = theta + thetastep;
+		#end
+		sphere { punkt(phi,theta), 0.01 }
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color Blue
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+#macro punkt2(a,b)
+	punkt(5.6+a*sqrt(3)/2-b/2,0.2+a/2 + b*sqrt(3)/2)
+#end
+
+#declare darkgreen = rgb<0,0.6,0>;
+
+#declare astart = 0;
+#declare aend = 1;
+#declare bstart = -0.2;
+#declare bend = 1.2;
+union {
+	#declare a = astart;
+	#declare astep = 0.2;
+	#while (a < aend + astep/2)
+		#declare b = bstart;
+		#declare bstep = (bend - bstart)/N;
+		#while (b < bend - bstep/2)
+			sphere { punkt2(a,b), 0.01 }
+			cylinder { punkt2(a,b), punkt2(a,b+bstep), 0.01 }
+			#declare b = b + bstep;
+		#end
+		sphere { punkt2(a,b), 0.01 }
+		#declare a = a + astep;
+	#end
+	pigment {
+		color darkgreen
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+union {
+	#declare b = bstart;
+	#declare bstep = 0.2;
+	#while (b < bend + bstep/2)
+		#declare a = astart;
+		#declare astep = (aend - astart)/N;
+		#while (a < aend - astep/2)
+			sphere { punkt2(a,b), 0.01 }
+			cylinder { punkt2(a,b), punkt2(a+astep,b), 0.01 }
+			#declare a = a + astep;
+		#end
+		sphere { punkt2(a,b), 0.01 }
+		#declare b = b + bstep;
+	#end
+	pigment {
+		color Orange
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}