From a331aefc04c6d615a693d892048e1615db86a99c Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Fri, 16 Apr 2021 01:01:31 +0200
Subject: Continue video script

---
 vorlesungen/punktgruppen/script.tex | 52 ++++++++++++++++++++++++-------------
 1 file changed, 34 insertions(+), 18 deletions(-)

(limited to 'vorlesungen/punktgruppen/script.tex')

diff --git a/vorlesungen/punktgruppen/script.tex b/vorlesungen/punktgruppen/script.tex
index a1e356a..2a6d95c 100644
--- a/vorlesungen/punktgruppen/script.tex
+++ b/vorlesungen/punktgruppen/script.tex
@@ -4,37 +4,52 @@
 \usepackage{xcolor}
 
 \newcommand{\scene}[1]{\noindent[ #1 ]\par}
-\newenvironment{totranslate}{\color{red!60!black}}{}
+\newenvironment{totranslate}{\color{blue!70!black}}{}
 
 \begin{document}
 
 \section{Intro}
 
 \section{Geometrie}
-\scene{Intro}
-\scene{Zyklische Gruppe}
-
 \begin{totranslate}
-  Let's now focus our attention on the simplest class of symmetries: those
-  generated by a single rotation. We describe the symmetries with a group
+We'll start with geometric symmetries as they are the simplest to grasp.
+
+\scene{Intro}
+  To mathematically formulate the concept, we will think of symmetries as
+  actions to perform on an object, like this square. The simplest action, is to
+  take this square, do nothing and put it back down. Another action could be to
+  flip it along an axis, or to rotate it around its center by 90 degrees.
+
+\scene{Cyclic Groups}
+  Let's focus our attention on the simplest class of symmetries: those
+  generated by a single rotation. We will gather the symmetries in a group
   \(G\), and denote that it is generated by a rotation \(r\) with these angle
   brackets.
   
-  Take this shape as an example. By applying the rotation \emph{action} 5
-  times, it looks as if we had not done anything, furthermore, if we \emph{act}
-  with higher ``powers'' \(r\), they will have the same effect as one of the
-  previous action.  Thus the group only contain the identity and the powers of
-  \(r\) up to 4.
+  Take this pentagon as an example. By applying the rotation \emph{action} 5
+  times, it is the same as if we had not done anything, furthermore, if we
+  \emph{act} a sixth time with \(r\), it will be the same as if we had just
+  acted with \(r\) once.  Thus the group only contain the identity and the
+  powers of \(r\) up to 4.
   
-  In general, groups with this structure are known as the
-  ``Cyclic Groups'' of order \(n\), where the action \(r\) can be applied
-  \(n-1\) times before wrapping around.
-\end{totranslate}
+  In general, groups with this structure are known as the ``Cyclic Groups'' of
+  order \(n\), where the action \(r\) can be applied \(n-1\) times before
+  wrapping around. 
 
-\scene{Diedergruppe}
+  % You can think of them as the rotational symmetries of an \(n\)-gon.
 
-\begin{totranslate}
-  Okay that was not difficult, now let's spice this up a bit. 
+\scene{Dihedral Groups}
+  Okay that was not difficult, now let's spice this up a bit. Consider this
+  group for a square, generated by two actions: a rotation \(r\) and a
+  reflection \(\sigma\). Because we have two actions we have to write in the
+  generator how they relate to each other.
+
+  Let's analyze this expression. Two reflections are the same as the identity.
+  Four rotations are the same as the identity, and a rotation followed by a
+  reflection, twice, is the same as the identity.
+
+  This forms a group with 8 possible unique actions. This too can be generalized
+  to an \(n\)-gon, and is known as the ``Dihedral Group'' of order \(n\).
 \end{totranslate}
 
 \scene{Symmetrische Gruppe}
@@ -45,3 +60,4 @@
 \section{Krystalle}
 
 \end{document}
+% vim:et ts=2 sw=2:
-- 
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