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\title{log(m) meeting 12}
%\date{\today}
\lhead{ Log(M): Origami on lattices \\}
\rhead{ Meeting 12 \\ 12 April 2019}
\begin{document}


\section{Logistics}

\ul{Next meeting}:  Tuesday April 16, 11:30 - 1pm
\medskip

    
\bigskip
{Expectations} for next meeting: 
\begin{itemize}
\item \textbf{[Final poster]}
    Create draft of  poster,
    using the template here: 
    \url{https://sites.lsa.umich.edu/logm/resources/}
    (final poster due \textbf{Thursday, April 18}!)
    
    Send me a copy by \textbf{Monday evening}, so I have time to look over it before our meeting Tuesday.


    \item $3$-fold distance formula
    
    In a hexagon suppose we choose fold angles $\theta_1,\theta_2,\theta_3$,
    and let $\alpha_{4,6}$ 
    denote the angle between crease vector $4$ and $6$
    (where $0\leq \alpha_{4,6}\leq \pi$).
    Then 
    \begin{align}
   \cos(\alpha_{4,6})
    &= \frac1{16}( 1 - 3\cos \theta_1 - 3\cos\theta_2 - 3\cos\theta_3 
    -3\cos\theta_1\cos\theta_2 \nonumber \\
    &\qquad \phantom{\frac1{16}} - 3\cos\theta_2\cos \theta_3 
    +9\cos\theta_1\cos\theta_3 -3\cos\theta_1\cos\theta_2\cos\theta_3 \nonumber \\
    &\qquad \phantom{\frac1{16}} + 6\sin\theta_1\sin\theta_2 + 6\sin\theta_2\sin\theta_3 + 6\cos\theta_1\sin\theta_2\sin\theta_3 \nonumber \\
    &\qquad \phantom{\frac1{16}} + 6\sin\theta_1\sin\theta_2\cos\theta_3 + 12\sin\theta_1\cos\theta_2\sin\theta_3)
\end{align}
    
    \textbf{Problem 12}:
    \begin{enumerate}[(a)]
    \item What is the $2$nd-order approximation of (1) 
    when fold angles $\theta_i$ are small?
    
    \item The answer to (a) is a constant plus a quadratic form in $\theta_1,\theta_2,\theta_3$.
    What is the signature of this quadratic form?
    
    \item If we impose the constraint
    \[ \cos(\alpha_{4,6}) = -\frac12 + \epsilon\]
    for small $\epsilon > 0$,
    then using the  approximation (a) what is the minimum possible value of 
    \[ E(\theta_1,\theta_2,\theta_3) = \theta_1^2 + \theta_2^2 + \theta_3^2 \]
    as a function of $\epsilon$?
    \end{enumerate}
    
     
    
    \item \textbf{[Writing]} 
    Write up notes for this meeting, and
    continue writing up relevant discussion from this week in draft of final report 
    
\end{itemize}

\section{Energy propagation}
Using our understanding of a single hexagon we would like to know
how energy ``propagates'' in a hexagon lattice.
We can reduce this to the following concrete question

\setcounter{prob}{12}
\begin{prob}
In a single hexagon, suppose we fix fold angle $6$ at some nonzero value $\theta_0 >0$.
The rest of the hexagon cannot lie completely flat.
We would like to quantify this as follows:
can we find some constant $C$ such that
\[ \max \{\theta_1,\theta_2,\ldots,\theta_5 \} \geq C\theta_0\]
for any fold configuration?
\end{prob}

\end{document}