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\documentclass[12pt, xcolor, aspectratio=169]{beamer}
% language
\usepackage{polyglossia}
\setmainlanguage{german}
% Theme
\beamertemplatenavigationsymbolsempty
% set look
\usetheme{default}
\usecolortheme{fly}
\usefonttheme{serif}
%% Set font
\usepackage[p,osf]{scholax}
\usepackage{amsmath}
\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}
% set colors
\definecolor{background}{HTML}{202020}
\setbeamercolor{normal text}{fg=white, bg=background}
\setbeamercolor{structure}{fg=white}
\setbeamercolor{item projected}{use=item,fg=background,bg=item.fg!35}
\setbeamercolor*{palette primary}{use=structure,fg=white,bg=structure.fg}
\setbeamercolor*{palette secondary}{use=structure,fg=white,bg=structure.fg!75}
\setbeamercolor*{palette tertiary}{use=structure,fg=white,bg=structure.fg!50}
\setbeamercolor*{palette quaternary}{fg=white,bg=background}
\setbeamercolor*{block title}{parent=structure}
\setbeamercolor*{block body}{fg=background, bg=}
\setbeamercolor*{framesubtitle}{fg=white}
\setbeamertemplate{section page}
{
\begin{center}
\Huge
\insertsection
\end{center}
}
\AtBeginSection{\frame{\sectionpage}}
% Metadata
\title{\LARGE \scshape Punktgruppen und Kristalle}
\author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz}
\institute{Hochschule f\"ur Technik OST, Rapperswil}
\date{10. Mai 2021}
% Slides
\begin{document}
\frame{\titlepage}
\frame{\tableofcontents}
\section{Einleitung}
\frame{
\[
\psi
\]
}
\section{Geometrische Symmetrien}
%% Made in video
\section{Algebraische Symmetrien}
\frame{
\begin{columns}
\begin{column}{.3\textwidth}
Produkt mit \(i\)
\begin{align*}
1 \cdot i &= i \\
i \cdot i &= -1 \\
-1 \cdot i &= -i \\
-i \cdot i &= 1
\end{align*}
\pause
%
Gruppe
\begin{align*}
G &= \left\{
1, i, -1, -i
\right\} \\
&= \left\{
1, i, i^2, i^3
\right\} \\
Z_4 &= \left\{
\mathbb{1}, r, r^2, r^3
\right\}
\end{align*}
\pause
%
\end{column}
\begin{column}{.5\textwidth}
%
Darstellung
\[
\phi : Z_4 \to G
\]
\begin{align*}
\phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\
\phi(r) &= i & \phi(r^3) &= i^3
\end{align*}
\pause
%
Homomorphismus
\begin{align*}
\phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\
&= i \cdot 1
\end{align*}
\pause
%
\(\phi\) ist bijektiv \(\implies Z_4 \cong G\)
\end{column}
\end{columns}
}
\section{Kristalle}
\section{Anwendungen}
\end{document}
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