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Diffstat (limited to 'vorlesungen/slides/3/multiplikation.tex')
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diff --git a/vorlesungen/slides/3/multiplikation.tex b/vorlesungen/slides/3/multiplikation.tex new file mode 100644 index 0000000..13f4e03 --- /dev/null +++ b/vorlesungen/slides/3/multiplikation.tex @@ -0,0 +1,180 @@ +% +% multiplikation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\N{21} +\begin{frame}[t,fragile] +\frametitle{Multiplikation mit $\alpha$ in $\mathbb{Z}(\alpha)$} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] + +\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\sqrt{2})$}; +\node[color=blue] at (3.5,3.2) [above left] {$\sqrt{2}\mathbb{Z}(\sqrt{2})$}; + +\pgfmathparse{sqrt(2)} +\xdef\a{\pgfmathresult} +\pgfmathparse{-int(3.2/\a)} +\xdef\ymin{\pgfmathresult} +\pgfmathparse{int(3.2/\a)} +\xdef\ymax{\pgfmathresult} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\sqrt{2}$}]; + +\def\punkt#1#2#3{ + ({(1-(#3))*(#1)+2*(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1))*\a}) +} + +\foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.5pt] + \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08]; + } +} +\foreach \y in {\ymin,...,\ymax}{ + \draw[color=red,line width=0.5pt] + \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0}; +} + + +\def\bildnetz#1{ + \pgfmathparse{(#1-1)/(\N-1)} + \xdef\t{\pgfmathresult} + \only<#1>{ + \uncover<2->{ + \draw[->,color=blue,line width=1.4pt] + (0,\a) -- \punkt{0}{1}{\t}; + \draw[->,color=blue,line width=1.4pt] + (1,0) -- \punkt{1}{0}{\t}; + } + \foreach \x in {-3,...,3}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=blue] + \punkt{\x}{\y}{\t} + circle[radius=0.06]; + } + } + \foreach \y in {\ymin,...,\ymax}{ + \draw[color=blue,line width=0.5pt] + \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t}; + } + } +} + +\begin{scope} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\ifthenelse{\boolean{presentation}}{ + \foreach \T in {1,...,\N}{ + \bildnetz{\T} + } +}{ + \bildnetz{\N} +} +\end{scope} + +\uncover<\N->{ +\begin{scope}[yshift=-2.5cm] +\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8); +\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8); +\node at (0,0) {$\displaystyle W=\begin{pmatrix}0&2\\1&0\end{pmatrix}$}; +\end{scope} +} + +\node at (0,-3.7) {$\alpha^2 = 2$}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] + +\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\varphi)$}; +\node[color=blue] at (3.5,3.2) [above left] {$\varphi\mathbb{Z}(\varphi)$}; + +\pgfmathparse{(sqrt(5)+1)/2} +\xdef\a{\pgfmathresult} +\pgfmathparse{-int(3.3/\a)} +\xdef\ymin{\pgfmathresult} +\pgfmathparse{int(3.3/\a)} +\xdef\ymax{\pgfmathresult} +\def\punkt#1#2#3{ + ({(1-(#3))*(#1)+(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1+#2))*\a}) +} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\varphi$}]; + +\foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.5pt] + \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08]; + } +} +\foreach \y in {\ymin,...,\ymax}{ + \draw[color=red,line width=0.5pt] + \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0}; +} + +\def\bildnetz#1{ + \pgfmathparse{(#1-1)/(\N-1)} + \xdef\t{\pgfmathresult} + \only<#1>{ + \uncover<2->{ + \draw[->,color=blue,line width=1.4pt] + (0,\a) -- \punkt{0}{1}{\t}; + \draw[->,color=blue,line width=1.4pt] + (1,0) -- \punkt{1}{0}{\t}; + } + \foreach \x in {-3,...,3}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=blue] \punkt{\x}{\y}{\t} + circle[radius=0.06]; + } + } + \foreach \y in {\ymin,...,\ymax}{ + \draw[color=blue,line width=0.5pt] + \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t}; + } + } +} + +\begin{scope} + +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\ifthenelse{\boolean{presentation}}{ + \foreach \T in {1,...,\N}{ + \bildnetz{\T} + } +}{ + \bildnetz{\N} +} +\end{scope} + +\uncover<\N->{ +\begin{scope}[yshift=-2.5cm] +\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8); +\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8); +\node at (0,0) {$\displaystyle \Phi=\begin{pmatrix}0&1\\1&1\end{pmatrix}$}; +\end{scope} +} + +\node at (0,-3.7) {$\alpha^2 = \alpha + 1$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup |