Vorlesung vom 28.11.2013 digitalisiert

documents/GeoTopo/GeoTopo.tex

@@ -36,6 +36,7 @@
 \usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns,fadings,decorations.pathreplacing}
 \usepackage{tqft}
 \usepackage{cleveref} % has to be after hyperref, ntheorem, amsthm
+\usepackage{xspace}   % for new commands; decides weather I want to insert a space after the command
 \usepackage{shortcuts}
 
 \usepackage{fancyhdr}

documents/GeoTopo/Kapitel2.tex

@@ -718,5 +718,146 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
     $\Rightarrow \chi(\Delta^n) = 1 \qed$
 \end{beweis}
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Mitschrieb vom 28.11.2013                                         %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{definition}
+    \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
+        \item Ein 1D-Simplizialkomplex heißt \textbf{Graph}\xindex{Graph}.
+        \item Ein Graph, der homöomorph zu $S^1$ ist, heißt \textbf{Kreis}\xindex{Kreis}.
+        \item Ein zusammenhängender Graph heißt \textbf{Baum}\xindex{Baum},
+              wenn er keinen Kreis enthält.
+    \end{enumerate}
+\end{definition}
+
+\begin{figure}[ht]
+    \centering
+    \subfloat[Dies wird häufig auch als Multigraph bezeichnet.]{
+        \parbox{4cm}{\centering\input{figures/topology-graph-simple.tex}}
+        \label{fig:topology-graph-simple}
+    }%
+    \subfloat[Planare Einbettung des Tetraeders]{
+        \parbox{4cm}{\centering\input{figures/topology-graph-tetraeder.tex}}
+        \label{fig:topology-graph-tetraeder}
+    }
+
+    \subfloat[$K_5$]{
+        \parbox{4cm}{\centering\input{figures/topology-graph-k-5.tex}}
+        \label{fig:k-5}
+    }%
+    \subfloat[$K_{3,3}$]{
+        \parbox{4cm}{\centering\input{figures/topology-graph-k-3-3.tex}}
+        \label{fig:k-3-3}
+    }%
+    \label{fig:graphen-beispiele}
+    \caption{Beispiele für Graphen}
+\end{figure}
+
+\begin{korollar}
+    Für jeden Baum $T$ gilt $\gamma(T) = 1$.
+\end{korollar}
+
+\begin{beweis}
+    Induktion über die Anzahl der Ecken.
+\end{beweis}
+
+\begin{korollar}
+    \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
+        \item Jeder zusammenhängende Graph $\Gamma$ enthält einen
+              Teilbaum $T$, der alle Ecken von $\Gamma$ enthält.%
+              \footnote{$T$ wird \enquote{Spannbaum} genannt.}
+        \item Ist $n = a_1(\Gamma) = a_1(T)$, so ist $\chi(\Gamma) = 1 - n$.
+    \end{enumerate}
+\end{korollar}
+
+\begin{beweis}\leavevmode
+    \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
+        \item Siehe \enquote{Algorithmus von Kruskal}.
+        \item $\begin{aligned}[t]\chi(\Gamma) &= a_0(\Gamma) - a_1(\Gamma)\\
+                                        &= a_0(\Gamma) - (n+a_1(T))\\
+                                        &= a_0(T) - a_1(T) - n\\
+                                        &= \chi(T) - n\\
+                                        &= 1-n
+              \end{aligned}$
+    \end{enumerate}
+\end{beweis}
+
+\begin{korollar}\label{kor:simplex-unterteilung}
+    Sei $\Delta$ ein $n$-Simplex und $x \in \Delta^\circ \subseteq \mdr^n$.
+    Sei $K$ der Simplizialkomplex, der aus $\Delta$ durch 
+    \enquote{Unterteilung} in $x$ entsteht. Dann ist $\chi(K) = \chi(\Delta) = 1$.
+\end{korollar}
+
+\begin{figure}[ht]
+    \centering
+    \subfloat[$K$]{
+        \parbox{4cm}{\centering\input{figures/topology-graph-tetraeder-area.tex}}
+        \label{fig:topology-simplizial-complex-k}
+    }%
+    \subfloat[$\Delta$, das aus $K$ durch Unterteilung entsteht]{
+        \parbox{4cm}{\centering\input{figures/topology-graph-tetraeder-area-2.tex}}
+        \label{fig:topology-simplizial-complex-k-division}
+    }%
+    \label{fig:korollar-beispiel}
+    \caption{Beispiel für Korollar~\ref{kor:simplex-unterteilung}.}
+\end{figure}
+
+\begin{beweis}
+    $\chi(K) = \chi(\Delta) - \underbrace{\underbrace{(-1)^n}_{n-\text{Simplex}} + \sum_{k=0}^n (-1)^k}_{(1+(-1))^{n+1}} = \chi(\Delta) \qed$
+\end{beweis}
+
+\begin{satz}[Eulersche Polyederformel]\xindex{Eulersche Polyederformel}
+    Sei $P$ ein konvexes Polyeder in $\mdr^3$, d.~h. $\partial P$ ist
+    ein 2-dimensionaler Simplizialkomplex, sodass gilt:
+    \[\forall x,y \in \partial P: [x,y] \subseteq P\]
+
+    Dann ist $\chi(\partial P) = 2$.
+\end{satz}
+
+\begin{beweis}\leavevmode
+    \begin{enumerate}[label=\arabic*)]
+        \item Die Aussage ist richtig für den Tetraeder.
+        \item \Obda{} sei $0 \in P$ und $P \subseteq \fB_1(0)$. Projeziere
+              $0P$ von $0$ aus auf $\partial \fB_1(0) = S^2$.
+              Erhalte Triangulierung von $S^2$.
+
+              \todo[inline]{Bild von rundem Wuerfel}
+        \item Sind $P_1$ und $P_2$ konvexe Polygone und $T_1, T_2$
+              die zugehörigen Triangulierungen von $S^2$, so gibt es 
+              eine eine Triangulierungen $T$, die sowohl um $T_1$ als
+              auch um $T_2$ Verfeinerung ist.
+
+              \todo[inline]{Komische Zeichung}
+
+              Nach Korollar~\ref{kor:simplex-unterteilung} ist
+              $\chi(\partial P_1) = \chi(T_1) = \chi(T) = \chi(T_2) = \chi(\partial P_2) = 2$.
+              Weil \obda{} $P_2$ ein Tetraeder ist.
+    \end{enumerate}
+\end{beweis}
+
+\begin{korollar}
+    Sei $K$ ein \todo{Warum in Klammern?}{(endlicher)} Simplizialkomplex mit Eckenmenge $V$
+    und $<$ eine Totalordnung auf $V$.
+
+    Für jedes $n=0, \dots, d=\dim(K)$ sei $A_n(K)$ die Menge der
+    $n$-Simplizes von $K$ und $C_n(K)$ der $\mdr$-Vektorraum mit
+    Basis $A_n(K)$, d.~h.
+    \[C_n(K) = \Set{\sum_{\sigma \in A_n(K)} c_\sigma \cdot \sigma | c_\sigma \in \mdr}\]
+
+    Sei $\sigma = \Delta(x_0, \dots, x_n) \in A_n(K)$, sodass 
+    $x_0 < x_1 < \dots < x_n$.
+
+    Für $i = 0, \dots, n$ sei $\partial_i \sigma := \Delta(x_0, \dots, \hat{x_i}, \dots, x_n)$
+    die $i$-te Seite von $\sigma$. Sei $d_\sigma = d_n \sigma := \sum_{i=0} (-1)^i \partial_i \sigma \in C_{n-1} (K)$
+    und $d: C_n(K) \rightarrow C_{n-1}(K)$ die dadurch definierte lineare
+    Abbildung.
+
+    Dann gilt: $d_{n-1} \circ d_n = 0$
+
+    \todo[inline]{Skizze von Dreieck}
+
+    $d_2 \sigma = e_1 - e_2 + e_3 = c - b - (c-a) + b - a = 0$
+\end{korollar}
+
 % Die Übungsaufgaben sollen ganz am Ende des Kapitels sein.
 \input{Kapitel2-UB}

documents/GeoTopo/figures/todo.tex

@@ -0,0 +1,3 @@
+\begin{tikzpicture}
+    \path (0,0)  edge [bend angle=10,bend right] node[label=TODO] {} (-1,-1.5);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-graph-k-3-3.tex

@@ -0,0 +1,10 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+
+    \foreach \x in {0,1,2}
+    \foreach \y in {0,1,2}{
+      \node (a)[point] at (\y,0) {};
+      \node (b)[point] at (\x,1) {};
+        \draw (a) -- (b);
+    }
+\end{tikzpicture}

documents/GeoTopo/figures/topology-graph-k-5.tex

@@ -0,0 +1,17 @@
+    \newcommand\n{5}
+    \begin{tikzpicture}
+        \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+
+        \begin{scope}[rotate=17]
+        %the multiplication with floats is not possible. Thus I split the loop in two.
+        \foreach \number in {1,...,\n}{
+            \node[point] (N-\number) at ({\number*(360/\n)}:1.5cm) {};
+        }
+
+        \foreach \number in {1,...,\n}{
+            \foreach \y in {1,...,\n}{
+                \draw (N-\number) -- (N-\y);
+            }
+        }
+        \end{scope}
+    \end{tikzpicture}

documents/GeoTopo/figures/topology-graph-simple.tex

@@ -0,0 +1,8 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point] at (0,0) {};
+    \node (b)[point] at (1,0) {};
+    \path (a.center) edge [bend left]  (b.center);
+    \path (a.center) edge              (b.center);
+    \path (a.center) edge [bend right] (b.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-graph-tetraeder-area-2.tex

@@ -0,0 +1,15 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (z)[point] at (0,0) {};
+    \node (a)[point] at (90:1cm) {};
+    \node (b)[point] at (210:1cm) {};
+    \node (c)[point] at (330:1cm) {};
+    \node (d)[point] at (10:1.5cm) {};
+    \path (z.center) edge (a.center);
+    \path (z.center) edge (b.center);
+    \path (z.center) edge (c.center);
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+
+    \draw (a.center) -- (d.center);
+    \draw (c.center) -- (d.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-graph-tetraeder-area.tex

@@ -0,0 +1,15 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (z)[point] at (0,0) {};
+    \node (a)[point] at (90:1cm) {};
+    \node (b)[point] at (210:1cm) {};
+    \node (c)[point] at (330:1cm) {};
+    \path (z.center) edge (a.center);
+    \path (z.center) edge (b.center);
+    \path (z.center) edge (c.center);
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+
+    \draw[pattern=north west lines] (a.center) -- (b.center) -- (z.center) --cycle;
+    \draw[pattern=dots] (b.center) -- (c.center) -- (z.center) --cycle;
+    \draw[pattern=crosshatch] (a.center) -- (c.center) -- (z.center) --cycle;
+\end{tikzpicture}

documents/GeoTopo/figures/topology-graph-tetraeder.tex

@@ -0,0 +1,11 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (z)[point] at (0,0) {};
+    \node (a)[point] at (90:1cm) {};
+    \node (b)[point] at (210:1cm) {};
+    \node (c)[point] at (330:1cm) {};
+    \path (z.center) edge (a.center);
+    \path (z.center) edge (b.center);
+    \path (z.center) edge (c.center);
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+\end{tikzpicture}

documents/GeoTopo/shortcuts.sty

@@ -68,7 +68,9 @@
 \def\GL{\ensuremath{\mathrm{GL}}}
 \newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}
 \newcommand\dcup{\mathbin{\dot{\cup}}}
-\newcommand\obda{o.~B.~d.~A.}
-\newcommand\Obda{o.~B.~d.~A.}
+
+%%%Text %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand\obda{o.~B.~d.~A.\xspace}
+\newcommand\Obda{O.~B.~d.~A.\xspace}
 
 

tikz/topology-graph-k-3-3/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-graph-k-3-3
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-graph-k-3-3/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-graph-k-3-3.png)

tikz/topology-graph-k-3-3/topology-graph-k-3-3.tex

@@ -0,0 +1,15 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+
+\begin{document}
+    \begin{tikzpicture}
+        \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+
+        \foreach \x in {0,1,2}
+        \foreach \y in {0,1,2}{
+          \node (a)[point] at (\y,0) {};
+          \node (b)[point] at (\x,1) {};
+            \draw (a) -- (b);
+        }
+    \end{tikzpicture}
+\end{document}

tikz/topology-graph-k-5/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-graph-k-5
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-graph-k-5/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-graph-k-5.png)

tikz/topology-graph-k-5/topology-graph-k-5.tex

@@ -0,0 +1,22 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+
+\begin{document}
+    \newcommand\n{5}
+    \begin{tikzpicture}
+        \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+
+        \begin{scope}[rotate=17]
+        %the multiplication with floats is not possible. Thus I split the loop in two.
+        \foreach \number in {1,...,\n}{
+            \node[point] (N-\number) at ({\number*(360/\n)}:1.5cm) {};
+        }
+
+        \foreach \number in {1,...,\n}{
+            \foreach \y in {1,...,\n}{
+                \draw (N-\number) -- (N-\y);
+            }
+        }
+        \end{scope}
+    \end{tikzpicture}
+\end{document}

tikz/topology-graph-simple/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-graph-simple
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-graph-simple/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-graph-simple.png)

tikz/topology-graph-simple/topology-graph-simple.tex

@@ -0,0 +1,13 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point] at (0,0) {};
+    \node (b)[point] at (1,0) {};
+    \path (a.center) edge [bend left]  (b.center);
+    \path (a.center) edge              (b.center);
+    \path (a.center) edge [bend right] (b.center);
+\end{tikzpicture}
+\end{document}

tikz/topology-graph-tetraeder-area-2/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-graph-tetraeder-area-2
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-graph-tetraeder-area-2/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-graph-tetraeder-area-2.png)

tikz/topology-graph-tetraeder-area-2/topology-graph-tetraeder-area-2.tex

@@ -0,0 +1,20 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (z)[point] at (0,0) {};
+    \node (a)[point] at (90:1cm) {};
+    \node (b)[point] at (210:1cm) {};
+    \node (c)[point] at (330:1cm) {};
+    \node (d)[point] at (10:1.5cm) {};
+    \path (z.center) edge (a.center);
+    \path (z.center) edge (b.center);
+    \path (z.center) edge (c.center);
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+
+    \draw (a.center) -- (d.center);
+    \draw (c.center) -- (d.center);
+\end{tikzpicture}
+\end{document}

tikz/topology-graph-tetraeder-area/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-graph-tetraeder-area
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-graph-tetraeder-area/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-graph-tetraeder-area.png)

tikz/topology-graph-tetraeder-area/topology-graph-tetraeder-area.tex

@@ -0,0 +1,21 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+\usetikzlibrary{patterns}
+
+\begin{document}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (z)[point] at (0,0) {};
+    \node (a)[point] at (90:1cm) {};
+    \node (b)[point] at (210:1cm) {};
+    \node (c)[point] at (330:1cm) {};
+    \path (z.center) edge (a.center);
+    \path (z.center) edge (b.center);
+    \path (z.center) edge (c.center);
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+
+    \draw[pattern=north west lines] (a.center) -- (b.center) -- (z.center) --cycle;
+    \draw[pattern=dots] (b.center) -- (c.center) -- (z.center) --cycle;
+    \draw[pattern=crosshatch] (a.center) -- (c.center) -- (z.center) --cycle;
+\end{tikzpicture}
+\end{document}

tikz/topology-graph-tetraeder/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-graph-tetraeder
+DELAY = 80
+DENSITY = 300
+WIDTH = 512
+
+make:
+	pdflatex $(SOURCE).tex -output-format=pdf
+	make clean
+
+clean:
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
+
+gif:
+	pdfcrop $(SOURCE).pdf
+	convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
+	make clean
+
+png:
+	make
+	make svg
+	inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
+
+transparentGif:
+	convert $(SOURCE).pdf -transparent white result.gif
+	make clean
+
+svg:
+	#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
+	pdf2svg $(SOURCE).pdf $(SOURCE).svg
+	# Necessary, as pdf2svg does not always create valid svgs:
+	inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg

tikz/topology-graph-tetraeder/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-graph-tetraeder.png)

tikz/topology-graph-tetraeder/topology-graph-tetraeder.tex

@@ -0,0 +1,16 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (z)[point] at (0,0) {};
+    \node (a)[point] at (90:1cm) {};
+    \node (b)[point] at (210:1cm) {};
+    \node (c)[point] at (330:1cm) {};
+    \path (z.center) edge (a.center);
+    \path (z.center) edge (b.center);
+    \path (z.center) edge (c.center);
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+\end{tikzpicture}
+\end{document}