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-rw-r--r--vorlesungen/punktgruppen/script.pdfbin22295 -> 25412 bytes
-rw-r--r--vorlesungen/punktgruppen/script.tex52
2 files changed, 34 insertions, 18 deletions
diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf
index 044a426..293b248 100644
--- a/vorlesungen/punktgruppen/script.pdf
+++ b/vorlesungen/punktgruppen/script.pdf
Binary files differ
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: