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\title{log(m) meeting 3}
%\date{\today}
\lhead{ Log(M): Origami on lattices \\}
\rhead{ Meeting 3 \\ 1 February 2019}
\begin{document}
%\maketitle 
%\begin{center}
%Notes, Fall 2017, assorted
%\end{center}

\section{Logistics}
- Writing workshop coming up $\rightarrow$ 
let me know if you need information from me

\ul{Next meeting}: Saturday February 9, 12 - 1pm.

Next next meeting: Friday February 15, 9:30 - 11am.

\bigskip
\ul{Expectations} for next meeting: 
\begin{itemize}
    \item Focus on Problem 4: what is the space of fold configuration for 3 creases through a hexagon?
    Try to get a complete answer (I don't know what the answer is to this!)
    
    \item \textbf{[Writing]} Write up answer to Problem 1(b) as discussed; include some figures of the moduli space.
    
    \item \textbf{[Visualization]} Can you write code to visualize a space of fold configurations?
    I.e. for Problem 1 or 2, display the moduli space on one side of the screen and as you move the cursor over this space, display the fold configuration on the other side of the screen.
    
    \item Please bring back the straw polygons next week!
\end{itemize}

\section{Moduli spaces}

\begin{prob}
(a) What is the moduli space of folding configurations for 
two parallel creases, parallel to opposite sides, 
which split the sheet into $3$ regions of equal area?
\begin{center}
    \includegraphics[scale=0.5]{parallel_equal_inv}
\end{center}

(b) What is the moduli space of folding configurations for two parallel creases, parallel to opposite sides, which are spaced arbitrarily?

(c) What  distance is needed between the creases for this moduli space to look like Example 2 (the ``far away'' case)? 
What happens as the distance separating the two creases approaches $0$?
\end{prob}

\ul{Discussion}: 
In the general case with crease spacing $r_1, d, r_2$, the boundary 
condition when the sheet intersects itself is described 
by the equation
\[ \tan \theta_2  = \frac{r_2 \sin \theta_1}{d - r_2 \cos \theta_1},\]
with certain restrictions on $\theta_1$ and $\theta_2$.

We looked at graphing this condition on \url{desmos.com} for some values of $d, r_1, r_2$.

% \setcounter{prob}{2}
% \begin{prob} \,[non-folding example of a moduli space]
% What is the moduli space of ``space quadrilaterals'' with fixed side lengths?
% E.g. if I cut four lengths of straw (with four different colors) and connect them in a loop with string, what are all possible configurations of this object 
% (up to rotations and translations in $3$-space)?

% (Thanks to Alex Wright for the straw polygons!)
% \end{prob}

\setcounter{prob}{3}
\begin{prob}
What is the space of folding configurations of a hexagon with creases along its three axes?
\begin{center}
    \includegraphics[scale=0.5]{hexagon_inv}
\end{center}
\end{prob}


\section{Energy}
\ul{Big question}: how do materials behave in the real world?

\noindent i.e. in physics language: what configurations  {\em minimize energy} under given constraints?

For example, an object acted on by gravity will rest at the lowest point and a spring will tend towards its equilibrium length.
We are concerned not only with understanding a moduli space of all possible configurations, but also figuring out which configurations are more likely to show up due to physical considerations.

In the case of a folding sheet, a sharper fold should (usually) correspond to higher energy. 
Thus we can assign an {\em energy function} to a folding configuration $\Phi$
by, for example,
\[ E(\Phi) = \sum_i |\theta_i| \ell_i \]
where $\theta_1, \theta_2, \ldots$ are the dihedral angles in the configuration $\Phi$ and $\ell_1, \ell_2, \ldots$ are the lengths of the corresponding creases
($\theta_i = 0$ indicates an unfolded crease.)
More generally, we could consider the energy function
\[ E_p(\Phi) = \left(\sum_i |\theta_i|^p \ell_i \right)^{1/p}.\]
Given a moduli space of fold configurations and an energy function, we would like to understand the relation between energy and other ``geometric'' measures of a configuration, e.g. how much space does it take up, how far away are the two farthest points, how far is it from being flat, etc.

\end{document}