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\begin{document}
\begin{center} {\Large \textbf{
Worksheet 1: Symmetries}}
\end{center}

\begin{enumerate}
\item How many symmetries do
$$\begin{tikzpicture}[baseline=1cm]
\foreach \x/\y in {0/0,2/0,0/2,2/2}
	\node (\x\y) at (\x,\y) {$\bullet$};
\draw (0,0) -- (2,0) -- (2,2) -- (0,2) -- (0,0);
\end{tikzpicture}\quad \text{and}\quad\begin{tikzpicture}[baseline=1cm]
\foreach \x/\y in {0/0,4/0,0/2,4/2}
	\node (\x\y) at (\x,\y) {$\bullet$};
\draw (0,0) -- (4,0) -- (4,2) -- (0,2) -- (0,0);
\end{tikzpicture}$$
have?  What makes two symmetries different?



\item For each of the shapes in 1, number the corners.  
\begin{enumerate}
\item Is there a way to view a symmetry as a function whose domain an codomain are the set of corners?



\item From this point of view, give an example of a function on the corners that is not a symmetry of the above objects.



\end{enumerate}
\item What might a symmetry of a deck of cards look like?
\item  Classify the symmetries of 
$$\begin{tikzpicture}[baseline=1cm]
\foreach \x in {0,45,90,135,180,225,270,315}
	\node (\x) at (\x:2cm) [inner sep=-1pt] {$\bullet$};
\draw (0) -- (45) -- (90) -- (135) -- (180) -- (225) -- (270) -- (315) -- (0);
\end{tikzpicture}\ .$$
(Also be sure to say what it means to classify!)
\item Lists some properties that a symmetry (as a function) should have.  
\end{enumerate}

\noindent\textbf{Write-up (due September 06, 2019):}  Write an introduction to mathematical symmetry. 
\begin{itemize}
\item What is a symmetry?  
\item What kinds of things can it be applied to?  
\item Illustrate with the example of problem 4.
\end{itemize}

\end{document}
