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\title{log(m) meeting 9}
%\date{\today}
\lhead{ Log(M): Origami on lattices \\}
\rhead{ Meeting 9 \\ 22 March 2019}
\begin{document}


\section{Logistics}

\ul{Next meeting}:  Friday March 29, 9:30 - 11am
\medskip

    
\bigskip
{Expectations} for next meeting: 
\begin{itemize}

    
    
    \item mountain / valley labellings
    for hexagon crease pattern
    
    \textbf{Problem 9}: 
    In a diagram with $6$ creases coming together at a vertex at equal angles,
the unit null cone near the flat space is two disjoint spheres.
We consider cutting up  these spheres into regions according to mountain/valley labellings.
\begin{enumerate}[(a)]
    \item How many zero-dimensional, one-dimensional, and two-dimensional regions are there on each sphere?
    (Answer: $9$, $24$, and $17$)
    
    \item What shapes are the two-dimensional regions? 
    How do they fit together? (i.e. draw out a diagram)
\end{enumerate}
    
    \item  area formula for spherical $n$-gons
    
    We showed in Problem 10 that
    \[ \text{Area} = \sum_{i=1}^n \alpha_i - (n-2)\pi 
    \qquad\text{or}\qquad
    \text{Area} = 2\pi - \sum_{i=1}^n (\pi - \alpha_i) ,\]
    at least for convex polygons,
    where $\alpha_i$ are the internal angles and 
    $\pi-\alpha_i$ are the external ``fold'' angles.
    
    \textbf{Problem 12}: How can we interpret the spherical area formula for a ``polygon'' whose boundary intersects itself?
    
    Read about the {\em winding number} here
    \url{https://en.wikipedia.org/wiki/Winding_number},
    in particular the ``Intuitive description'' and ``Turning number'' subsections.
    
    
    \item angle constraints on hexagon crease pattern
    
    \textbf{Problem 11}:
    Suppose we want to fix three consecutive creases on a hexagon at fold angles
    $\theta_1$, $\theta_2$, and $\theta_3$. 
    (By ``fold angle'' we mean flat $\Leftrightarrow \, \theta_i=0$.)
    \begin{enumerate}[(a)]
    \item When is this possible, and when is this impossible?
    \item When this is possible, what are the values of angles $\theta_4,\, \theta_5,\, \theta_6$?
    Note: there may be more than one answer. How many answers are possible?
    \end{enumerate}
    Particular sub-problem:  
    
    \textbf{Problem 11 (c)}
     Suppose $\theta_1 = \theta_3 = 0$ and $\theta_2$ is arbitrary.
    Then one possibility is that $\theta_4 = \theta_6 = 0$
    and $\theta_5 = \theta_2$.
    
    A second possibility is that crease number $5$ is ``flipped inward'',
    so that $\theta_5 = -\theta_2$ and $\theta_4$ and $\theta_6$
    are at some positive angle. 
    What is the angle of $\theta_4= \theta_6$ as a function of $\theta_2$, in this case?
    
    
    
    \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}




\end{document}