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Vorlesung vom 26.11.2013 digitalisiert

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documents/GeoTopo/GeoTopo.pdf
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documents/GeoTopo/Kapitel2.tex
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@ -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}

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@ -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}

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\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}

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\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) {};
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\node (c)[point] at (2,0) {};
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\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);
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\draw (h.center) -- node[above] {$b$} (i.center);
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\draw (d.center) -- node[left] {$d$} (a.center);
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\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;
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\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) {};
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\node (g)[point] at (0,2) {};
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\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) {};
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\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}

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\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) {};
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\node (f)[point] at (4,2) {};
\node (g)[point] at (5,3) {};
\node (h)[point] at (1,3) {};
\end{tikzpicture}

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\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}

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\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}

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\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) {};
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\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}

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\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}

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\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}

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@ -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) {};
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\end{scope}
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\draw[pattern=north east lines] (a.center) -- (p.center) -- (b.center) -- cycle;
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\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}

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\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}

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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

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Compiled example
----------------
![Example](topology-1-d-simplizialkomplex.png)

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\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}

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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

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Compiled example
----------------
![Example](topology-2.png)

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\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}

31
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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

3
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Compiled example
----------------
![Example](topology-cube-divided.png)

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\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}

31
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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

3
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Compiled example
----------------
![Example](topology-cube.png)

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21
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\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}

31
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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

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

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23
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\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}

31
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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

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

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22
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\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}

31
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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

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

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17
tikz/topology-pyramid/topology-pyramid.tex Normal file
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\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}

31
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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

3
tikz/topology-triangle-no-simplicial-complex/Readme.md Normal file
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Compiled example
----------------
![Example](topology-triangle-no-simplicial-complex.png)

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24
tikz/topology-triangle-no-simplicial-complex/topology-triangle-no-simplicial-complex.tex Normal file
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\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}

31
tikz/topology-triangle-on-line/Makefile Normal file
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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

3
tikz/topology-triangle-on-line/Readme.md Normal file
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@ -0,0 +1,3 @@
Compiled example
----------------
![Example](topology-triangle-on-line.png)

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28
tikz/topology-triangle-on-line/topology-triangle-on-line.tex Normal file
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\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}

31
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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

3
tikz/topology-triangle-simplicial-complex/Readme.md Normal file
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@ -0,0 +1,3 @@
Compiled example
----------------
![Example](topology-triangle-simplicial-complex.png)

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26
tikz/topology-triangle-simplicial-complex/topology-triangle-simplicial-complex.tex Normal file
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\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}