Vorlesung vom 26.11.2013 digitalisiert

documents/GeoTopo/Kapitel2.tex

@@ -600,5 +600,135 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
     \caption{Beispiele für $k$-Simplexe}
 \end{figure}
 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Mitschrieb vom 21.11.2013                                         %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{definition}
+    \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
+        \item Eine endliche Menge $K$ von Simplizes im $\mdr^n$
+              heißt (endlicher) \textbf{Simplizialkomplex}\xindex{Simplizialkomplex},
+              wenn gilt:
+            \begin{enumerate}[label=(\roman*)]
+                \item Für $\Delta \in K$ und $S \subseteq \Delta$ Teilsimplex
+                      ist $S \in K$
+                \item Für $\Delta_1, \Delta_2 \in K$ ist 
+                      $\Delta_1 \cap \Delta_2$ leer
+                      oder ein Teilsimplex von $\Delta_1$ und von 
+                      $\Delta_2$
+            \end{enumerate}
+        \item $|K| := \bigcup_{\Delta \in K} \Delta$ (mit Spurtoplogie)
+              heißt \textbf{geometrische Realisierung}\xindex{Realisierung!geometrische}
+              von $K$.
+        \item Ist $d = \max \Set{ k | K \text{ enthält } k-\text{Simplex}}$,
+              so heißt $d$ \textbf{Dimension}\xindex{Dimension} von
+              $K$.
+    \end{enumerate}
+\end{definition}
+
+\xindex{Oktaeder}\xindex{Würfel}
+\begin{figure}[ht]
+    \centering
+    \subfloat[1D Simplizialkomplex]{
+        \parbox{4cm}{\centering\input{figures/topology-1-d-simplizialkomplex}}
+        \label{fig:simplizialkomplex-1-d}
+    }%
+    \subfloat[2D Simplizialkomplex (ohne untere Fläche!)]{
+        \parbox{4cm}{\centering\input{figures/topology-pyramid.tex}}
+        \label{fig:simplizialkomplex-2-d}
+    }%
+    \subfloat[2D Simplizialkomplex]{
+        \parbox{5cm}{\centering\input{figures/topology-oktaeder.tex}}
+        \label{fig:simplizialkomplex-2-d-okateder}
+    }%
+
+    \subfloat[1D Simplizialkomplex]{
+        \parbox{5cm}{\centering\input{figures/topology-cube.tex}}
+        \label{fig:simplizialkomplex-cube}
+    }%
+    \subfloat[2D Simplizialkomplex]{
+        \parbox{5cm}{\centering\input{figures/topology-cube-divided.tex}}
+        \label{fig:simplizialkomplex-cube-divided}
+    }
+
+    \subfloat[$P$ ist kein Teilsimplex, da Eigenschaft (ii) verletzt ist]{
+        \parbox{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}}
+        \label{fig:no-simplizialkomplex-triangles}
+    }%
+    \subfloat[Simplizialkomplex]{
+        \parbox{5cm}{\centering\input{figures/topology-triangle-simplicial-complex.tex}}
+        \label{fig:simplizialkomplex-triangles}
+    }%
+    \label{fig:simplizialkomplexe}
+    \caption{Beispiele für Simplizialkomplexe}
+\end{figure}
+
+\begin{definition}
+    Seien $K, L$ Simplizialkomplexe. Eine stetige Abbildung
+    \[f:|K| \rightarrow |L|\]
+    heißt \textbf{simplizial}\xindex{Abbildung!simpliziale}, wenn für
+    jedes $\Delta \in K$ gilt:
+    \begin{enumerate}[label=(\roman*)]
+        \item $f(\Delta) \in L$
+        \item $f|_{\Delta} : \Delta \rightarrow f(\Delta)$ ist eine
+              affine Abbildung.
+    \end{enumerate}
+\end{definition}
+
+\begin{beispiel}
+    \begin{enumerate}[label=\arabic*)]
+        \item $\varphi(e_1) := b_1$, $\varphi(e_2) := b_2$\\
+              $\varphi$ ist eine eindeutig bestimmte lineare Abbildung
+
+              \input{figures/topology-linear-mapping.tex}
+
+        \item Folgende Abbildung $\Delta^n \rightarrow \Delta^{n-1}$ 
+              ist simplizial:
+
+              \input{figures/topology-triangle-to-line.tex}
+        \item \todo[inline]{wozu dient das Beispiel?}
+
+            \input{figures/topology-2.tex}
+    \end{enumerate}
+\end{beispiel}
+
+\begin{definition}
+    Sei $K$ ein endlicher Simplizialkomplex. Für $n \geq 0$ sei
+    $a_n(K)$ die Anzahl der $n$-Simplizes in $K$.
+
+    Dann heißt 
+    \[\chi(K) := \sum_{k=0}^{\dim K} (-1)^n a_n(K)\]
+    \textbf{Eulerzahl}\xindex{Eulerzahl} (oder Euler-Charakteristik\index{Euler-Charakteristik|see{Eulerzahl}})
+    von $K$.
+\end{definition}
+
+\begin{beispiel}
+    \begin{enumerate}[label=\arabic*)]
+        \item $\chi(\Delta^1) = 2 - 1 = 1$\\
+              $\chi(\Delta^2) = 3 - 3 + 1 = 1$\\
+              $\chi(\Delta^3) = 4 - 6 + 4 - 1 = 1$
+        \item $\chi(\text{Oktaeder-Oberfläche}) = 6 - 12 + 8 = 2$\\
+              $\chi(\text{Rand des Tetraeders}) = 2$\\
+              $\chi(\text{Ikosaeder}) = 12 - 30 + 20 = 2$
+        \item $\chi(\text{Würfel}) = 8 - 12 + 6 = 2$\\
+              $\chi(\text{Würfel, unterteilt in Dreiecksflächen}) = 8 - (12 + 6) + (6 \cdot 2) = 2$
+    \end{enumerate}
+\end{beispiel}
+
+\begin{korollar}
+    $\chi(\Delta^n) = 1$ für jedes $n \in \mdn_0$
+\end{korollar}
+
+\begin{beweis}
+    $\Delta^n$ ist die konvexe Hülle von $(e_0, \dots, e_n)$ in $\mdr^{n+1}$.
+    Jede $(k+1)$-elementige Teilmenge von $\Set{e_0, \dots, e_n}$
+    definiert ein $k$-Simplex.\\
+    $\Rightarrow a_k(\Delta^n) = \binom{n+1}{k+1}, \;\;\; k = 0, \dots, n$\\
+    $\Rightarrow \chi(\Delta^n) = \sum_{k=0}^n (-1)^k \binom{n+1}{k+1}$\\
+    $f(x) = (x+1)^{n+1} \stackrel{\substack{\tiny\text{Binomischer}\\\text{Lehrsatz}}}{=} \sum_{k=0}^{n+1} \binom{n+1}{k} x^k$\\
+    $\Rightarrow 0 = \sum_{k=0}^{n+1} \binom{n+1}{k} (-1)^k = \chi(\Delta^n) -1$\\
+    $\Rightarrow \chi(\Delta^n) = 1 \qed$
+\end{beweis}
+
 % Die Übungsaufgaben sollen ganz am Ende des Kapitels sein.
 \input{Kapitel2-UB}

documents/GeoTopo/Symbolverzeichnis.tex

@@ -257,6 +257,13 @@
   sort=ZZZRang
 }
 
+\newglossaryentry{Einschraeunkung}
+{
+  name={$f|_U$},
+  description={$f$ eingeschränkt auf U},
+  sort=ZZZEinschraenkung
+}
+
 % Setze den richtigen Namen für das Glossar
 \renewcommand*{\glossaryname}{\glossarName}
 \deftranslation{Glossary}{\glossarName}

documents/GeoTopo/figures/topology-1-d-simplizialkomplex.tex

@@ -0,0 +1,10 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+    \node (a)[point] at (0.4,0) {};
+    \node (b)[point] at (1,1) {};
+    \node (c)[point] at (2,1) {};
+    \node (d)[point] at (2.6,0) {};
+    \node (e)[point] at (2,-1) {};
+    \node (f)[point] at (1,-1) {};
+    \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- (e.center) -- (f.center) -- cycle;
+\end{tikzpicture}

documents/GeoTopo/figures/topology-2.tex

@@ -0,0 +1,78 @@
+\begin{tikzpicture}[scale=1.5]
+    \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) {};
+    \node (c)[point] at (2,0) {};
+
+    \node (d)[point] at (0,1) {};
+    \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {};
+    \node (f)[point] at (2,1) {};
+
+    \node (g)[point] at (0,2) {};
+    \node (h)[point] at (1,2) {};
+    \node (i)[point] at (2,2) {};
+
+    \draw (a.center) -- node[below] {$a$} (b.center);
+    \draw (b.center) -- node[below] {$b$} (c.center);
+    \draw (g.center) -- node[above] {$a$} (h.center);
+    \draw (h.center) -- node[above] {$b$} (i.center);
+    \draw (d.center) -- node[left] {$c$} (g.center);
+    \draw (d.center) -- node[left] {$d$} (a.center);
+    \draw (f.center) -- node[right] {$d$} (c.center);
+    \draw (i.center) -- node[right] {$c$} (f.center);
+
+    \draw (a.center) -- (b.center) -- (M.center) -- (d.center) -- cycle;
+    \draw (b.center) -- (c.center) -- (f.center) -- (M.center) -- cycle;
+    \draw (d.center) -- (M.center) -- (h.center) -- (g.center) -- cycle;
+    \draw (M.center) -- (f.center) -- (i.center) -- (h.center) -- cycle;
+
+    \draw[thick, blue] (d.center) -- (M.center) -- (f.center);
+
+    \draw[thick, red] (a.center) -- (b.center);
+    \draw[thick, red] (h.center) -- (i.center);
+
+    % Draw again so that lines are below nodes
+    \draw (a.center) -- (i.center);
+    \draw (c.center) -- (g.center);
+
+    \node (a)[point] at (0,0) {};
+    \node (b)[point] at (1,0) {};
+    \node (c)[point] at (2,0) {};
+
+    \node (d)[point] at (0,1) {};
+    \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {};
+    \node (f)[point] at (2,1) {};
+
+    \node (g)[point] at (0,2) {};
+    \node (h)[point] at (1,2) {};
+    \node (i)[point] at (2,2) {};
+
+    \draw[->] (3,1) -- node[above] {Quotient nach} node[below] {Punktspiegelung} (5,1);
+
+    \begin{scope}[xshift=6.5cm,yshift=1cm]
+        \node (w)[point] at (-1,0) {};
+        \node (x)[point] at (0,-1) {};
+        \node (y)[point] at (1,0) {};
+        \node (z)[point] at (0,1) {};
+        \node (m)[point] at (0,0) {};
+
+        \node (1) at (-0.3,+0.35) {1};
+        \node (2) at (+0.3,+0.35) {2};
+        \node (3) at (+0.3,-0.35) {3};
+        \node (4) at (-0.3,-0.35) {4};
+
+        \draw (w.center) -- (x.center) -- (y.center) -- (z.center) -- cycle;
+
+        \draw[thick, blue] (w.center) -- (m.center);
+        \draw (m.center) -- (y.center);
+        \draw (x.center) -- (m.center) -- (z.center);
+        \draw[thick, red] (z.center) -- (y.center) -- (x.center);
+
+        % Draw again, as lines should be below points
+        \node (w)[point] at (-1,0) {};
+        \node (x)[point] at (0,-1) {};
+        \node (y)[point] at (1,0) {};
+        \node (z)[point] at (0,1) {};
+        \node (m)[point] at (0,0) {};
+    \end{scope}
+\end{tikzpicture}

documents/GeoTopo/figures/topology-cube-divided.tex

@@ -0,0 +1,30 @@
+\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 (4,0) {};
+    \node (c)[point] at (5,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \node (g)[point] at (5,3) {};
+    \node (h)[point] at (1,3) {};
+    \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle;
+    \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle;
+    \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle;
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (h.center);
+
+    \draw[orange] (b.center) -- (e.center) -- (g.center);
+    \draw[orange,dashed] (a.center) -- (c.center) -- (h.center);
+    \draw[orange,dashed] (d.center) -- (e.center);
+    \draw[orange] (f.center) -- (c.center);
+
+    \node (a)[point] at (0,0) {};
+    \node (b)[point] at (4,0) {};
+    \node (c)[point] at (5,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \node (g)[point] at (5,3) {};
+    \node (h)[point] at (1,3) {};
+\end{tikzpicture}

documents/GeoTopo/figures/topology-cube.tex

@@ -0,0 +1,16 @@
+\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 (4,0) {};
+    \node (c)[point] at (5,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \node (g)[point] at (5,3) {};
+    \node (h)[point] at (1,3) {};
+    \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle;
+    \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle;
+    \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle;
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (h.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-linear-mapping.tex

@@ -0,0 +1,18 @@
+\begin{tikzpicture}
+    \tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
+
+    \node (a)[point,label={[label distance=0cm]180:$0$}] at (0,0) {};
+    \node (b)[point,label={[label distance=0cm]0:$e_2$}] at (2.5,0) {};
+    \node (c)[point,label={[label distance=0cm]90:$e_1$}] at (2,1) {};
+
+    \begin{scope}[xshift=5cm]
+        \node (d)[point,label={[label distance=0cm]180:$0$}] at (0,0) {};
+        \node (e)[point,label={[label distance=0cm]0:$b_1$}] at (4,0) {};
+        \node (f)[point,label={[label distance=0cm]90:$b_2$}] at (2,2) {};
+    \end{scope}
+
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+    \draw (d.center) -- (e.center) -- (f.center) -- cycle;
+
+    \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (5,0.5);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-oktaeder.tex

@@ -0,0 +1,17 @@
+\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 (2,0) {};
+    \node (c)[point] at (3,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (1.5,3) {};
+    \node (f)[point] at (1.5,-2) {};
+    \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center);
+    \draw (a.center) -- (e.center);
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (e.center);
+
+    \draw (a.center) -- (f.center) -- (b.center);
+    \draw (f.center) -- (c.center);
+    \draw[dashed] (f.center) -- (d.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-pyramid.tex

@@ -0,0 +1,12 @@
+\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 (2,0) {};
+    \node (c)[point] at (3,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (1.5,3) {};
+    \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center);
+    \draw (a.center) -- (e.center);
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (e.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-triangle-no-simplicial-complex.tex

@@ -0,0 +1,18 @@
+\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 (3,0) {};
+    \node (c)[point] at (2,2) {};
+
+    \begin{scope}[yshift=2cm]
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \end{scope}
+    
+    \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {};
+
+    \draw[pattern=north east lines] (a.center) -- (b.center) -- (c.center) -- cycle;
+    \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle;
+    \draw (p.center) -- (d.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-triangle-simplicial-complex.tex

@@ -0,0 +1,20 @@
+\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 (3,0) {};
+    \node (c)[point] at (2,2) {};
+
+    \begin{scope}[yshift=2cm]
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \end{scope}
+    
+    \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {};
+
+    \draw[pattern=north east lines] (a.center) -- (p.center) -- (b.center) -- cycle;
+    \draw[pattern=north west lines] (a.center) -- (p.center) -- (c.center) -- cycle;
+    \draw[pattern=vertical lines]   (b.center) -- (p.center) -- (c.center) -- cycle;
+    \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle;
+    \draw (p.center) -- (d.center);
+\end{tikzpicture}

documents/GeoTopo/figures/topology-triangle-to-line.tex

@@ -0,0 +1,23 @@
+\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 (2.5,0) {};
+    \node (c)[point] at (2,1) {};
+
+    \begin{scope}[xshift=4.5cm, yshift=0.5cm]
+        \node (d)[point] at (0,0) {};
+        \node (e)[point] at (2,0) {};
+    \end{scope}
+
+    \begin{scope}[xshift=1.5cm,yshift=0.6cm,rotate=-55]
+    \draw[->] (0,-0.1) -- (0.4,-0.1);
+    \draw[->] (0, 0.0) -- (0.5, 0.0);
+    \draw[->] (0, 0.1) -- (0.6, 0.1);
+    \end{scope}
+
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+    \draw (d.center) -- (e.center);
+
+    \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (4,0.5);
+\end{tikzpicture}

tikz/topology-1-d-simplizialkomplex/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-1-d-simplizialkomplex
+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-1-d-simplizialkomplex/Readme.md

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

tikz/topology-1-d-simplizialkomplex/topology-1-d-simplizialkomplex.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]
+    \node (a)[point] at (0.4,0) {};
+    \node (b)[point] at (1,1) {};
+    \node (c)[point] at (2,1) {};
+    \node (d)[point] at (2.6,0) {};
+    \node (e)[point] at (2,-1) {};
+    \node (f)[point] at (1,-1) {};
+    \draw (a.center) -- (b.center) -- (c.center) -- (d.center) -- (e.center) -- (f.center) -- cycle;
+\end{tikzpicture}
+\end{document}

tikz/topology-2/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-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-2/Readme.md

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

tikz/topology-2/topology-2.tex

@@ -0,0 +1,83 @@
+\documentclass[varwidth=true, border=2pt]{standalone}
+\usepackage{tikz}
+
+\begin{document}
+\begin{tikzpicture}[scale=1.5]
+    \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) {};
+    \node (c)[point] at (2,0) {};
+
+    \node (d)[point] at (0,1) {};
+    \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {};
+    \node (f)[point] at (2,1) {};
+
+    \node (g)[point] at (0,2) {};
+    \node (h)[point] at (1,2) {};
+    \node (i)[point] at (2,2) {};
+
+    \draw (a.center) -- node[below] {$a$} (b.center);
+    \draw (b.center) -- node[below] {$b$} (c.center);
+    \draw (g.center) -- node[above] {$a$} (h.center);
+    \draw (h.center) -- node[above] {$b$} (i.center);
+    \draw (d.center) -- node[left] {$c$} (g.center);
+    \draw (d.center) -- node[left] {$d$} (a.center);
+    \draw (f.center) -- node[right] {$d$} (c.center);
+    \draw (i.center) -- node[right] {$c$} (f.center);
+
+    \draw (a.center) -- (b.center) -- (M.center) -- (d.center) -- cycle;
+    \draw (b.center) -- (c.center) -- (f.center) -- (M.center) -- cycle;
+    \draw (d.center) -- (M.center) -- (h.center) -- (g.center) -- cycle;
+    \draw (M.center) -- (f.center) -- (i.center) -- (h.center) -- cycle;
+
+    \draw[thick, blue] (d.center) -- (M.center) -- (f.center);
+
+    \draw[thick, red] (a.center) -- (b.center);
+    \draw[thick, red] (h.center) -- (i.center);
+
+    % Draw again so that lines are below nodes
+    \draw (a.center) -- (i.center);
+    \draw (c.center) -- (g.center);
+
+    \node (a)[point] at (0,0) {};
+    \node (b)[point] at (1,0) {};
+    \node (c)[point] at (2,0) {};
+
+    \node (d)[point] at (0,1) {};
+    \node (M)[point,label={[label distance=0cm]5:$M$}] at (1,1) {};
+    \node (f)[point] at (2,1) {};
+
+    \node (g)[point] at (0,2) {};
+    \node (h)[point] at (1,2) {};
+    \node (i)[point] at (2,2) {};
+
+    \draw[->] (3,1) -- node[above] {Quotient nach} node[below] {Punktspiegelung} (5,1);
+
+    \begin{scope}[xshift=6.5cm,yshift=1cm]
+        \node (w)[point] at (-1,0) {};
+        \node (x)[point] at (0,-1) {};
+        \node (y)[point] at (1,0) {};
+        \node (z)[point] at (0,1) {};
+        \node (m)[point] at (0,0) {};
+
+        \node (1) at (-0.3,+0.35) {1};
+        \node (2) at (+0.3,+0.35) {2};
+        \node (3) at (+0.3,-0.35) {3};
+        \node (4) at (-0.3,-0.35) {4};
+
+        \draw (w.center) -- (x.center) -- (y.center) -- (z.center) -- cycle;
+
+        \draw[thick, blue] (w.center) -- (m.center);
+        \draw (m.center) -- (y.center);
+        \draw (x.center) -- (m.center) -- (z.center);
+        \draw[thick, red] (z.center) -- (y.center) -- (x.center);
+
+        % Draw again, as lines should be below points
+        \node (w)[point] at (-1,0) {};
+        \node (x)[point] at (0,-1) {};
+        \node (y)[point] at (1,0) {};
+        \node (z)[point] at (0,1) {};
+        \node (m)[point] at (0,0) {};
+    \end{scope}
+\end{tikzpicture}
+\end{document}

tikz/topology-cube-divided/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-cube-divided
+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-cube-divided/Readme.md

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

tikz/topology-cube-divided/topology-cube-divided.tex

@@ -0,0 +1,35 @@
+\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 (4,0) {};
+    \node (c)[point] at (5,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \node (g)[point] at (5,3) {};
+    \node (h)[point] at (1,3) {};
+    \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle;
+    \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle;
+    \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle;
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (h.center);
+
+    \draw[orange] (b.center) -- (e.center) -- (g.center);
+    \draw[orange,dashed] (a.center) -- (c.center) -- (h.center);
+    \draw[orange,dashed] (d.center) -- (e.center);
+    \draw[orange] (f.center) -- (c.center);
+
+    \node (a)[point] at (0,0) {};
+    \node (b)[point] at (4,0) {};
+    \node (c)[point] at (5,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \node (g)[point] at (5,3) {};
+    \node (h)[point] at (1,3) {};
+\end{tikzpicture}
+\end{document}

tikz/topology-cube/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-cube
+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-cube/Readme.md

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

tikz/topology-cube/topology-cube.tex

@@ -0,0 +1,21 @@
+\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 (4,0) {};
+    \node (c)[point] at (5,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \node (g)[point] at (5,3) {};
+    \node (h)[point] at (1,3) {};
+    \draw (a.center) -- (b.center) -- (f.center) -- (e.center) -- cycle;
+    \draw (b.center) -- (c.center) -- (g.center) -- (f.center) -- cycle;
+    \draw (e.center) -- (f.center) -- (g.center) -- (h.center) -- cycle;
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (h.center);
+\end{tikzpicture}
+\end{document}

tikz/topology-linear-mapping/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-linear-mapping
+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-linear-mapping/Readme.md

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

tikz/topology-linear-mapping/topology-linear-mapping.tex

@@ -0,0 +1,23 @@
+\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,label={[label distance=0cm]180:$0$}] at (0,0) {};
+    \node (b)[point,label={[label distance=0cm]0:$e_2$}] at (2.5,0) {};
+    \node (c)[point,label={[label distance=0cm]90:$e_1$}] at (2,1) {};
+
+    \begin{scope}[xshift=5cm]
+        \node (d)[point,label={[label distance=0cm]180:$0$}] at (0,0) {};
+        \node (e)[point,label={[label distance=0cm]0:$b_1$}] at (4,0) {};
+        \node (f)[point,label={[label distance=0cm]90:$b_2$}] at (2,2) {};
+    \end{scope}
+
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+    \draw (d.center) -- (e.center) -- (f.center) -- cycle;
+
+    \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (5,0.5);
+\end{tikzpicture}
+\end{document}

tikz/topology-oktaeder/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-oktaeder
+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-oktaeder/Readme.md

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

tikz/topology-oktaeder/topology-oktaeder.tex

@@ -0,0 +1,22 @@
+\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 (2,0) {};
+    \node (c)[point] at (3,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (1.5,3) {};
+    \node (f)[point] at (1.5,-2) {};
+    \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center);
+    \draw (a.center) -- (e.center);
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (e.center);
+
+    \draw (a.center) -- (f.center) -- (b.center);
+    \draw (f.center) -- (c.center);
+    \draw[dashed] (f.center) -- (d.center);
+\end{tikzpicture}
+\end{document}

tikz/topology-pyramid/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-pyramid
+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-pyramid/Readme.md

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

tikz/topology-pyramid/topology-pyramid.tex

@@ -0,0 +1,17 @@
+\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 (2,0) {};
+    \node (c)[point] at (3,1) {};
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (1.5,3) {};
+    \draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center);
+    \draw (a.center) -- (e.center);
+    \draw[dashed] (a.center) -- (d.center) -- (c.center);
+    \draw[dashed] (d.center) -- (e.center);
+\end{tikzpicture}
+\end{document}

tikz/topology-triangle-no-simplicial-complex/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-triangle-no-simplicial-complex
+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-triangle-no-simplicial-complex/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-triangle-no-simplicial-complex.png)

tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.tex

@@ -0,0 +1,24 @@
+\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 (a)[point] at (0,0) {};
+    \node (b)[point] at (3,0) {};
+    \node (c)[point] at (2,2) {};
+
+    \begin{scope}[yshift=2cm]
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \end{scope}
+    
+    \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {};
+
+    \draw[pattern=north east lines] (a.center) -- (b.center) -- (c.center) -- cycle;
+    \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle;
+    \draw (p.center) -- (d.center);
+\end{tikzpicture}
+\end{document}

tikz/topology-triangle-on-line/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-triangle-on-line
+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-triangle-on-line/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-triangle-on-line.png)

tikz/topology-triangle-on-line/topology-triangle-on-line.tex

@@ -0,0 +1,28 @@
+\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 (2.5,0) {};
+    \node (c)[point] at (2,1) {};
+
+    \begin{scope}[xshift=4.5cm, yshift=0.5cm]
+        \node (d)[point] at (0,0) {};
+        \node (e)[point] at (2,0) {};
+    \end{scope}
+
+    \begin{scope}[xshift=1.5cm,yshift=0.6cm,rotate=-55]
+    \draw[->] (0,-0.1) -- (0.4,-0.1);
+    \draw[->] (0, 0.0) -- (0.5, 0.0);
+    \draw[->] (0, 0.1) -- (0.6, 0.1);
+    \end{scope}
+
+    \draw (a.center) -- (b.center) -- (c.center) -- cycle;
+    \draw (d.center) -- (e.center);
+
+    \draw[thick,->] (3,0.5) -- node[above] {$\varphi$} (4,0.5);
+\end{tikzpicture}
+\end{document}

tikz/topology-triangle-simplicial-complex/Makefile

@@ -0,0 +1,31 @@
+SOURCE = topology-triangle-simplicial-complex
+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-triangle-simplicial-complex/Readme.md

@@ -0,0 +1,3 @@
+Compiled example
+----------------
+![Example](topology-triangle-simplicial-complex.png)

tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.tex

@@ -0,0 +1,26 @@
+\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 (a)[point] at (0,0) {};
+    \node (b)[point] at (3,0) {};
+    \node (c)[point] at (2,2) {};
+
+    \begin{scope}[yshift=2cm]
+    \node (d)[point] at (1,1) {};
+    \node (e)[point] at (0,2) {};
+    \node (f)[point] at (4,2) {};
+    \end{scope}
+    
+    \node (p)[point,label={[label distance=0cm]5:$P$}] at (1.5,0.5) {};
+
+    \draw[pattern=north east lines] (a.center) -- (p.center) -- (b.center) -- cycle;
+    \draw[pattern=north west lines] (a.center) -- (p.center) -- (c.center) -- cycle;
+    \draw[pattern=vertical lines]   (b.center) -- (p.center) -- (c.center) -- cycle;
+    \draw[pattern=dots] (d.center) -- (e.center) -- (f.center) -- cycle;
+    \draw (p.center) -- (d.center);
+\end{tikzpicture}
+\end{document}