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| -rw-r--r-- | vorlesungen/punktgruppen/script.pdf | bin | 22295 -> 25412 bytes | |||
| -rw-r--r-- | vorlesungen/punktgruppen/script.tex | 52 |
2 files changed, 34 insertions, 18 deletions
diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf Binary files differindex 044a426..293b248 100644 --- a/vorlesungen/punktgruppen/script.pdf +++ b/vorlesungen/punktgruppen/script.pdf 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: |