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\title{General Mathematics and Computational Science II}
\author{Midterm Exam}
\date{March 14, 2012}
%\date{March 23, 2012}
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\begin{enumerate}

\item Show that the complete graph with $5$ vertices cannot be
embedded in the plane.
\hfill(10)

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\item Show that a finite graph is bipartite if and only if it does not
contain a cycle of odd length.

(Recall that a graph is bipartite if its vertex set can be partitioned
into two disjoint subsets $V_1$ and $V_2$ such that every edge has one
vertex in $V_1$ and the other vertex in $V_2$.)
\hfill(5+5)
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\item 
\begin{enumerate}
\item Given three concentric circles, how can you construct an
equilateral triangle with one vertex on each of the circles?

\item Give a necessary and sufficient condition for the existence of
such an equilateral triangle.
\end{enumerate}
\hfill(5+5)

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\includegraphics[width=5cm]{concentric}
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\item Consider a transformation of the plane written in vector form as
\[
  F(\boldsymbol{v}) = \mathsf M \boldsymbol v
  \qquad \text{where} \qquad
  \mathsf M = 
  \begin{pmatrix}
    a & b \\ c & d
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item
Show that $F$ is a motion of the plane (i.e., preserves lengths) if
and only if
\begin{gather*}
  a^2 + c^2 = 1 \,, \\
  b^2 + d^2 = 1 \,, \\
  ab + cd = 0 \,.
\end{gather*}
\item What is the geometric significance of these conditions when you
consider the vectors
\[
  \boldsymbol u = 
  \begin{pmatrix}
  a \\ c
  \end{pmatrix}
  \qquad \text{and} \qquad
  \boldsymbol w = 
  \begin{pmatrix}
    b \\ d
  \end{pmatrix} \,?
\]
\end{enumerate}
\hfill(5+5)
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\item When $G$ is a group, the cyclic group generated by some $a \in
G$ is called a \emph{group cycle}.  Now construct a graph whose
vertices are the elements of $G$.  Insert an edge whenever two of the
group elements are adjacent in one of the group cycles (ordered
naturally).  This graph is known as the \emph{cycle
graph}.\footnote{The usual definition restricts to \emph{primitive
cycles}, those which do not appear as a subset of another cycle, but
this shall not matter here.}

\begin{enumerate}
\item Prove that the cycle graph of a finite group is connected.
\item Draw the cycle graph for the dihedral group
\[
  D_3 = 
  \{ \langle a, b \rangle \colon a^3 = e, b^2 = e, ab=ba^{-1} \} \,.
\]

\end{enumerate}
\hfill(5+5)
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\item Suppose the symmetry group $G$ of an ornament contains $H_0$,
the point reflection about the origin, and the glide reflection $U$,
chosen such that $U^2$ is the translation by one unit along the
$x$-axis.

Show that $G$ must contain point reflections about all points
$(n/2,0)$ with $n \in \mathbb Z$ and line reflections about all lines
$x=1/4+n/2$ with $n \in \mathbb Z$.  \hfill(10)
\end{enumerate}
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