Add pauses and minor fixes

4 files changed, 173 insertions, 156 deletions

diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf
index 6b17967..70ea683 100644
--- a/vorlesungen/punktgruppen/script.pdf
+++ b/vorlesungen/punktgruppen/script.pdf
diff --git a/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt b/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt
index 37d8b30..7275d5c 100644
--- a/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt
+++ b/vorlesungen/punktgruppen/shotcut/Punktgruppen/Punktgruppen.mlt
@@ -1,52 +1,24 @@
 <?xml version="1.0" standalone="no"?>
 <mlt LC_NUMERIC="C" version="7.0.0" title="Shotcut version 21.05.01" producer="main_bin">
   <profile description="HD 1080p 25 fps" width="1920" height="1080" progressive="1" sample_aspect_num="1" sample_aspect_den="1" display_aspect_num="16" display_aspect_den="9" frame_rate_num="60" frame_rate_den="1" colorspace="709"/>
-  <chain id="chain0" out="00:01:05.850">
-    <property name="length">00:01:05.867</property>
+  <chain id="chain0" out="00:01:09.983">
+    <property name="length">4200</property>
     <property name="eof">pause</property>
-    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/crystals/480p15/AlgebraicSymmetries.mp4</property>
+    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/camera/DSCF6392.MOV</property>
     <property name="mlt_service">avformat-novalidate</property>
     <property name="seekable">1</property>
-    <property name="audio_index">-1</property>
-    <property name="video_index">0</property>
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-    <property name="shotcut:hash">cbe611893baad5bff7cbdb23a3d4e7f3</property>
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-    <property name="creation_time">2021-05-05T11:17:04</property>
-    <property name="xml">was here</property>
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-  <chain id="chain1" out="00:01:39.050">
-    <property name="length">5944</property>
-    <property name="eof">pause</property>
-    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/crystals/480p15/Geometric2DSymmetries.mp4</property>
-    <property name="mlt_service">avformat-novalidate</property>
-    <property name="seekable">1</property>
-    <property name="audio_index">-1</property>
-    <property name="video_index">0</property>
-    <property name="mute_on_pause">0</property>
-    <property name="xml">was here</property>
-    <property name="shotcut:hash">47ac11813561d23004945f1738115b32</property>
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-  <chain id="chain2" out="00:00:43.850">
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-    <property name="eof">pause</property>
-    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/crystals/480p15/Geometric3DSymmetries.mp4</property>
-    <property name="mlt_service">avformat-novalidate</property>
-    <property name="seekable">1</property>
-    <property name="audio_index">-1</property>
+    <property name="audio_index">1</property>
     <property name="video_index">0</property>
     <property name="mute_on_pause">0</property>
     <property name="xml">was here</property>
-    <property name="shotcut:hash">f5af3911322b3cb8d68246387eef0a48</property>
+    <property name="shotcut:hash">d255c7f03c50cb9dcfb2f9e0db73cb05</property>
   </chain>
   <playlist id="main_bin">
     <property name="xml_retain">1</property>
-    <entry producer="chain0" in="00:00:00.000" out="00:01:05.850"/>
-    <entry producer="chain1" in="00:00:00.000" out="00:01:39.050"/>
-    <entry producer="chain2" in="00:00:00.000" out="00:00:43.850"/>
+    <entry producer="chain0" in="00:00:00.000" out="00:01:09.983"/>
   </playlist>
-  <producer id="black" in="00:00:00.000" out="00:02:22.917">
-    <property name="length">00:02:22.933</property>
+  <producer id="black" in="00:00:00.000" out="00:01:09.983">
+    <property name="length">00:01:10.000</property>
     <property name="eof">pause</property>
     <property name="resource">0</property>
     <property name="aspect_ratio">1</property>
@@ -55,47 +27,54 @@
     <property name="set.test_audio">0</property>
   </producer>
   <playlist id="background">
-    <entry producer="black" in="00:00:00.000" out="00:02:22.917"/>
+    <entry producer="black" in="00:00:00.000" out="00:01:09.983"/>
   </playlist>
-  <chain id="chain3" out="00:01:39.050">
-    <property name="length">5944</property>
+  <chain id="chain1" out="00:01:09.983">
+    <property name="length">4200</property>
     <property name="eof">pause</property>
-    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/crystals/480p15/Geometric2DSymmetries.mp4</property>
+    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/camera/DSCF6392.MOV</property>
     <property name="mlt_service">avformat-novalidate</property>
     <property name="seekable">1</property>
     <property name="audio_index">-1</property>
     <property name="video_index">0</property>
     <property name="mute_on_pause">0</property>
     <property name="xml">was here</property>
-    <property name="shotcut:hash">47ac11813561d23004945f1738115b32</property>
-    <property name="shotcut:caption">Geometric2DSymmetries.mp4</property>
+    <property name="shotcut:hash">d255c7f03c50cb9dcfb2f9e0db73cb05</property>
+    <property name="shotcut:caption">DSCF6392.MOV</property>
+    <property name="shotcut:defaultAudioIndex">1</property>
   </chain>
-  <chain id="chain4" out="00:00:43.850">
-    <property name="length">2632</property>
+  <playlist id="playlist0">
+    <property name="shotcut:video">1</property>
+    <property name="shotcut:name">V1</property>
+    <entry producer="chain1" in="00:00:00.000" out="00:01:09.983"/>
+  </playlist>
+  <chain id="chain2" out="00:01:09.983">
+    <property name="length">4200</property>
     <property name="eof">pause</property>
-    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/crystals/480p15/Geometric3DSymmetries.mp4</property>
+    <property name="resource">/home/npross/docs/school/hsr/MathSem1/SeminarMatrizen/vorlesungen/punktgruppen/media/videos/camera/DSCF6392.MOV</property>
     <property name="mlt_service">avformat-novalidate</property>
     <property name="seekable">1</property>
-    <property name="audio_index">-1</property>
+    <property name="audio_index">1</property>
     <property name="video_index">0</property>
     <property name="mute_on_pause">0</property>
     <property name="xml">was here</property>
-    <property name="shotcut:hash">f5af3911322b3cb8d68246387eef0a48</property>
-    <property name="shotcut:caption">Geometric3DSymmetries.mp4</property>
+    <property name="shotcut:hash">d255c7f03c50cb9dcfb2f9e0db73cb05</property>
+    <property name="shotcut:caption">DSCF6392.MOV</property>
+    <property name="shotcut:defaultAudioIndex">1</property>
   </chain>
-  <playlist id="playlist0">
-    <property name="shotcut:video">1</property>
-    <property name="shotcut:name">V1</property>
-    <entry producer="chain3" in="00:00:00.000" out="00:01:39.050"/>
-    <entry producer="chain4" in="00:00:00.000" out="00:00:43.850"/>
+  <playlist id="playlist1">
+    <property name="shotcut:audio">1</property>
+    <property name="shotcut:name">A1</property>
+    <entry producer="chain2" in="00:00:00.000" out="00:01:09.983"/>
   </playlist>
-  <tractor id="tractor0" title="Shotcut version 21.05.01" in="00:00:00.000" out="00:02:22.917">
+  <tractor id="tractor0" title="Shotcut version 21.05.01" in="00:00:00.000" out="00:01:09.983">
     <property name="shotcut">1</property>
-    <property name="shotcut:scaleFactor">0.124449</property>
+    <property name="shotcut:scaleFactor">0.174581</property>
     <property name="shotcut:projectAudioChannels">2</property>
     <property name="shotcut:projectFolder">1</property>
     <track producer="background"/>
     <track producer="playlist0"/>
+    <track producer="playlist1" hide="video"/>
     <transition id="transition0">
       <property name="a_track">0</property>
       <property name="b_track">1</property>
@@ -111,5 +90,12 @@
       <property name="threads">0</property>
       <property name="disable">1</property>
     </transition>
+    <transition id="transition2">
+      <property name="a_track">0</property>
+      <property name="b_track">2</property>
+      <property name="mlt_service">mix</property>
+      <property name="always_active">1</property>
+      <property name="sum">1</property>
+    </transition>
   </tractor>
 </mlt>
diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf
index 45ae1dd..37ada17 100644
--- a/vorlesungen/punktgruppen/slides.pdf
+++ b/vorlesungen/punktgruppen/slides.pdf
diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex
index d38065d..cc62969 100644
--- a/vorlesungen/punktgruppen/slides.tex
+++ b/vorlesungen/punktgruppen/slides.tex
@@ -64,12 +64,11 @@
 \frame{\titlepage}
 \frame{\tableofcontents}
 
-\section{Einleitung}
 \frame{
   \begin{itemize}
-    \item Was heisst \emph{Symmetrie} in der Mathematik?
-    \item Wie kann ein Kristall modelliert werden?
-    \item Aus der Physik: Piezoelektrizit\"at
+    \item Was heisst \emph{Symmetrie} in der Mathematik? \pause
+    \item Wie kann ein Kristall modelliert werden? \pause
+    \item Aus der Physik: Piezoelektrizit\"at \pause
   \end{itemize}
   \begin{center}
     \begin{tikzpicture}
@@ -125,7 +124,7 @@
         \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {};
         \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {};
 
-        \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U \neq 0\)} (n);
+        \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n);
       \end{scope}
     \end{tikzpicture}
   \end{center}
@@ -148,16 +147,19 @@
       \[
         G = \left\{\mathbb{1}, r, \sigma, \dots \right\}
       \]
+      \pause
       Matrixdarstellung
       \begin{align*}
         \Phi : G &\to O(3) \\
         g &\mapsto \Phi_g
       \end{align*}
+      \pause
       Orthogonale Gruppe
       \[
         O(n) = \left\{ Q : QQ^t = Q^tQ = I \right\}
       \]
     \end{column}
+    \pause
     \begin{column}{.5\textwidth}
       \begin{align*}
         \Phi_\mathbb{1} &= \begin{pmatrix}
@@ -181,15 +183,31 @@
 }
 
 \section{Kristalle}
+\begin{frame}[fragile]{M\"ogliche Kristallstrukturen}
+  \begin{center}
+    \begin{tikzpicture}[]
+      \node[circle, dashed, draw = gray, 
+        thick, fill = background,
+        minimum size = 4cm] {};
+      \node[gray] at (.9,-1.2) {674};
+
+      \node[circle, draw = white, thick,
+        fill = orange!40!background,
+        xshift = -3mm, yshift = 2mm,
+        minimum size = 2.75cm] (A) {};
+      \node[white, yshift = 2mm] at (A) {230};
+
+      \node[circle, draw = white, thick, 
+        fill = red!20!background,
+        xshift = -5mm, yshift = -5mm,
+        minimum size = 1cm] {32};
+    \end{tikzpicture}
+  \end{center}
+\end{frame}
+
 \begin{frame}[fragile]{}
-  \begin{columns}[T]
+  \begin{columns}
     \begin{column}{.5\textwidth}
-      Kristallgitter:
-      \(n_i \in \mathbb{Z}\),
-      \(\vec{a}_i \in \mathbb{R}^3\)
-      \[
-        \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3
-      \]
       \begin{center}
           \begin{tikzpicture}[
               dot/.style = {
@@ -201,14 +219,14 @@
             ]
 
             \begin{scope}
-              \clip (-1,-1) rectangle (4,3);
-              \foreach \y in {-5,-4,...,5} {
-                \foreach \x in {-5,-4,...,5} {
+              \clip (-2,-2) rectangle (3,4);
+              \foreach \y in {-7,-6,...,7} {
+                \foreach \x in {-7,-6,...,7} {
                   \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {};
                 }
               }
             \end{scope}
-            \draw[white, thick] (-1, -1) rectangle (4,3);
+            \draw[white, thick] (-2, -2) rectangle (3,4);
 
             \draw[red!80!background, thick, ->] 
               (N00) to node[midway, below] {\(\vec{a}_1\)} (N10);
@@ -217,31 +235,21 @@
           \end{tikzpicture}
       \end{center}
     \end{column}
+    \pause
     \begin{column}{.5\textwidth}
-      Invariant (symmetrisch) unter Translation
+      Kristallgitter:
+      \(n_i \in \mathbb{Z}\),
+      \(\vec{a}_i \in \mathbb{R}^3\)
+      \[
+        \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3
+      \]
+      \vspace{1cm}
+      \pause
+      
+      Invariant unter Translation
       \[
         Q_i(\vec{r}) = \vec{r} + \vec{a}_i
       \]
-	    M\"ogliche Kristallstrukturen
-      \begin{center}
-        \begin{tikzpicture}[]
-          \node[circle, dashed, draw = gray, 
-            thick, fill = background,
-            minimum size = 4cm] {};
-          \node[gray] at (.9,-1.2) {674};
-
-          \node[circle, draw = white, thick,
-            fill = orange!40!background,
-            xshift = -3mm, yshift = 2mm,
-            minimum size = 2.75cm] (A) {};
-          \node[white, yshift = 2mm] at (A) {230};
-
-          \node[circle, draw = white, thick, 
-            fill = red!20!background,
-            xshift = -5mm, yshift = -5mm,
-            minimum size = 1cm] {32};
-        \end{tikzpicture}
-      \end{center}
     \end{column}
   \end{columns}
 \end{frame}
@@ -249,7 +257,9 @@
 \begin{frame}[fragile]{}
   \begin{columns}[T]
     \begin{column}{.5\textwidth}
-      Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien?
+      \onslide<1->{
+        Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien?
+      }
       \begin{center}
         \begin{tikzpicture}[
             dot/.style = {
@@ -260,50 +270,70 @@
             },
           ]
 
-          \node[dot] (A1) at (0,0) {};
-          \node[below left] at (A1) {\(A\)};
-
-          \node[dot] (A2) at (2.5,0) {};
-          \node[below right] at (A2) {\(A'\)};
-
-          \draw[red!80!background, thick, ->] 
-            (A1) to node[midway, below] {\(\vec{Q}\)} (A2);
-
-          \node[dot] (B1) at (120:2.5) {};
-          \node[above left] at (B1) {\(B\)};
-
-          \draw[green!70!background, thick, ->]
-            (A1) ++(.5,0) arc (0:120:.5) node[midway, above, xshift=1mm] {\(C_n\)};
-          \draw[red!80!background, dashed, thick, ->] (A1) to (B1);
+          \onslide<2->{
+            \node[dot] (A1) at (0,0) {};
+            \node[below left] at (A1) {\(A\)};
+          }
 
+          \onslide<3->{
+            \node[dot] (A2) at (2.5,0) {};
+            \node[below right] at (A2) {\(A'\)};
 
-          \node[dot] (B2) at ($(A2)+(60:2.5)$) {};
-          \node[above right] at (B2) {\(B'\)};
-
-          \draw[green!70!background, thick, dashed, ->] (A2) ++(-.5,0) arc (180:60:.5);
-          \draw[red!80!background, dashed, thick, ->] (A2) to (B2);
-
-          \draw[yellow!80!background, thick, ->] (B1) to node[above, midway] {\(\vec{Q}'\)} (B2);
-
+            \draw[red!80!background, thick, ->] 
+              (A1) to node[midway, below] {\(\vec{Q}\)} (A2);
+          }
+
+          \onslide<4->{
+            \node[dot] (B1) at (120:2.5) {};
+            \node[above left] at (B1) {\(B\)};
+
+            \draw[green!70!background, thick, ->]
+              (A1) ++(.5,0) arc (0:120:.5)
+              node[midway, above, xshift=1mm] {\(C_n\)};
+            \draw[red!80!background, dashed, thick, ->] (A1) to (B1);
+          }
+
+          \onslide<5->{
+            \node[dot] (B2) at ($(A2)+(60:2.5)$) {};
+            \node[above right] at (B2) {\(B'\)};
+
+            \draw[green!70!background, thick, dashed, ->]
+              (A2) ++(-.5,0) arc (180:60:.5);
+            \draw[red!80!background, dashed, thick, ->] (A2) to (B2);
+          }
+
+          \onslide<6->{
+            \draw[yellow!80!background, thick, ->]
+              (B1) to node[above, midway] {\(\vec{Q}'\)} (B2);
+          }
+
+          \onslide<7->{
           \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (X) {};
           \draw[gray, dashed, thick] (A2) to (A2 |- B2);
+          }
 
-          \node[above left, xshift=-2mm] at (X) {\(x\)};
+          \onslide<8->{
+            \node[above left, xshift=-2mm] at (X) {\(x\)};
+          }
         \end{tikzpicture}
       \end{center}
     \end{column}
     \begin{column}{.5\textwidth}
-      Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\)
-      \begin{align*}
-        q' = n q &= q + 2x \\
-        nq &= q + 2q\sin(\alpha - \pi/2) \\
-        n &= 1 - 2\cos\alpha
-      \end{align*}
-      Somit muss
+      \onslide<9->{
+        Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\)
+      }
       \begin{align*}
-        \alpha &= \cos^{-1}\left(\frac{n-1}{2}\right) \\[1em]
-        \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\}
+        \onslide<10->{q' = n q &= q + 2x \\}
+        \onslide<11->{nq &= q + 2q\sin(\alpha - \pi/2) \\}
+        \onslide<12->{n &= 1 - 2\cos\alpha}
       \end{align*}
+      \onslide<13->{
+        Somit muss
+        \begin{align*}
+          \alpha &= \cos^{-1}\left(\frac{1-n}{2}\right) \\[1em]
+          \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\}
+        \end{align*}
+      }
     \end{column}
   \end{columns}
 \end{frame}
@@ -343,8 +373,9 @@
       negative/.style = { fill = blue!50 },
     ]
 
-    \node[font = {\large\bfseries}, align = center] (title) at (6,0) {Mit und Ohne\\ Symmetriezentrum};
+    \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum};
     \node[below = of title] {Polarisation Feld \(\vec{E}_p\)};
+    \pause
 
     \begin{scope}
       \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] {
@@ -354,6 +385,7 @@
       };
       \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
     \end{scope}
+    \pause
 
     \begin{scope}[yshift=-4.5cm]
       \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] {
@@ -372,6 +404,7 @@
 
       \draw[gray, dashed] (W) to (N) to (E) to (S) to (W);
     \end{scope}
+    \pause
 
     \begin{scope}[xshift=11cm]
       \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} {
@@ -379,27 +412,23 @@
       }
 
       \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
-      \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
+      % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
     \end{scope}
+    \pause
 
-    \begin{scope}[xshift=6cm, yshift=-4.5cm]
-      \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
-      \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
-      \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
-      \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
-      \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
-      \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
+    \begin{scope}[xshift=11cm, yshift=-4.5cm]
+      \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
+      \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
+      \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
+      \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
+      \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
+      \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
 
       \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
       % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
 
-      \foreach \d in {C1, C2} {
-        \draw[orange, very thick, <-] (\d) to ++(0,.7);
-      }
-
-      \foreach \d in {C4, C5} {
-        \draw[orange, very thick, <-] (\d) to ++(0,-.7);
-      }
+      \draw[orange, very thick, <-] (C6) to ++(.7,0);
+      \draw[orange, very thick, <-] (C3) to ++(-.7,0);
 
       \node[white] (E) {\(\vec{E}_p\)};
       \begin{scope}[node distance = .5mm]
@@ -409,20 +438,26 @@
       \draw[gray, thick, dotted] (E) to ++(0,2);
       \draw[gray, thick, dotted] (E) to ++(0,-2);
     \end{scope}
+    \pause
 
-    \begin{scope}[xshift=11cm, yshift=-4.5cm]
-      \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {};
-      \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {};
-      \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {};
-      \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {};
-      \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {};
-      \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {};
+    \begin{scope}[xshift=5.5cm, yshift=-4.5cm]
+      \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {};
+      \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {};
+      \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {};
+      \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {};
+      \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {};
+      \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {};
 
       \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1);
       % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2);
 
-      \draw[orange, very thick, <-] (C6) to ++(.7,0);
-      \draw[orange, very thick, <-] (C3) to ++(-.7,0);
+      \foreach \d in {C1, C2} {
+        \draw[orange, very thick, <-] (\d) to ++(0,.7);
+      }
+
+      \foreach \d in {C4, C5} {
+        \draw[orange, very thick, <-] (\d) to ++(0,-.7);
+      }
 
       \node[white] (E) {\(\vec{E}_p\)};
       \begin{scope}[node distance = .5mm]
@@ -462,15 +497,11 @@
       \]
       Anisotropisch Dielektrikum
       \[
-        (\ten{K}\ten{\varepsilon})\vec{E} = \frac{\omega^2}{\mu k^2} \vec{E}
+        (\ten{K}\ten{\varepsilon})\vec{E} = \frac{k^2}{\mu \omega^2} \vec{E}
       \]
       \[
         \vec{E} \in U_\lambda \implies (\ten{K}\ten{\varepsilon}) \vec{E} = \lambda \vec{E}
       \]
-      \"Ahenlich auch in der Mechanik
-      \[
-        \vec{F} = \kappa \vec{x} \quad \text{(Hooke)}
-      \]
     \end{column}
   \end{columns}
 }