Merge branch 'master' into master

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diff --git a/buch/chapters/60-gruppen/images/karten.tex b/buch/chapters/60-gruppen/images/karten.tex
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+++ b/buch/chapters/60-gruppen/images/karten.tex
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+%
+% 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}
+