\documentclass{article}
\usepackage{amssymb, amsmath, amsfonts, amsthm, graphics, enumerate, mathrsfs, mathtools, tikz-cd}
\usepackage{geometry, soul, hyperref}
\usetikzlibrary{positioning,arrows,scopes}


\usepackage{fancyhdr}
\pagestyle{fancy}

% \setlength{\headheight}{26pt}
% \setlength{\oddsidemargin}{-0.2in}    
% \setlength{\evensidemargin}{-0.2in}
% \setlength{\topmargin}{0in}
% \setlength{\textwidth}{6.5in}
 %\setlength{\headwidth}{6.5in} 
 \setlength{\textheight}{8.5in}
 
\theoremstyle{definition}
\newtheorem*{thm}{Theorem}
\newtheorem*{pthm}{Pre-Theorem}
\newtheorem{prob}{Problem}
\newtheorem{ex}{Exercise}
\newtheorem*{dfn}{Definition}
\newtheorem{eg}{Example}
\newtheorem*{prop}{Proposition}
\newtheorem*{rmk}{Remark}
\newtheorem*{conj}{Conjecture}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cM}{\mathcal{M}}

\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\indeg}{indeg}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\im}{im}

\newcommand{\dsp}{\displaystyle}
\newcommand{\floor}[1]{\lfloor{{#1}}\rfloor}
\newcommand{\ceil}[1]{\lceil{{#1}}\rceil}

\title{log(m) meeting 7}
%\date{\today}
\lhead{ Log(M): Origami on lattices \\}
\rhead{ Meeting 7 \\ 1 March 2019}
\begin{document}


\section{Logistics}

\ul{Next meeting}:  Friday March 15, 9:30 - 11am
\medskip

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

    
    
    \item Problem 9: mountain / valley labellings
    for hexagon crease pattern
    
    \item Problem 10: using the formula for spherical triangles relating external angles to (spherical) area,
    prove the generalization to spherical $n$-gons.
    
    
    \item \textbf{[Writing]} 
    Continue writing up relevant discussion from this week in draft of final report 
    

\end{itemize}

\section{Hexagon configuration space}

% \subsection{Changing coordinates}
% \setcounter{prob}{6}
% \begin{prob}
% Given three nonzero vectors $v_0,\,v_1,\,v_2 \in \RR^3$, 
% what is a formula to express the dihedral angle between 
% the planes $p_{01} = \RR v_0+\RR v_1$ and $p_{12} = \RR v_1 + \RR v_2$?

% \noindent (Possible hint: \url{https://en.wikipedia.org/wiki/Dihedral_angle#Mathematical_background})
% \end{prob}
% Our goal will be to use this expression for the angles to express the {\em energy} 
% \[ E(\Phi) = \left( \sum_i |\theta_i - \pi|^2 \right)^{1/2}\]
% of a given fold configuration in terms of the crease vectors,
% and to study configurations of constant energy.

\subsection{Mountain / valley diagrams}
To understand a topological space it often helps to cut it up into smaller, more manageable pieces.

For a fold configuration which is ``near flat,'' 
we say a crease is a {\em mountain fold} if it is higher than a secant line between its two adjacent flat regions, and a {\em valley fold} if it is lower.
(We take the nearby flat configuration as reference for what ``higher''
and ``lower'' mean.)

Suppose we label each crease in a configuration with ``mountain'' or ``valley'' or neither.
Visually we can indicate these respectively by a solid line, a dotted line, or no line.
\begin{eg}
For two perpendicular creases (so there are $4$ total crease vectors),
the following shows possible mountain/valley labellings:
\begin{center}
    \includegraphics[scale=0.3]{mv_diagram_small}
\end{center}
The following mountain/valley labelling is not possible:
%\begin{center}
\qquad    \includegraphics[scale=0.1]{mv_impossible}
%\end{center}
\end{eg}

\setcounter{prob}{7}
\begin{prob}
In the following diagrams with $5$ creases coming from a central vertex,
which mountain/valley labellings are possible? 
How do these fit together in configuration space near the unfolded state?
\begin{enumerate}[(a)]
    \item creases with equal spacings \qquad 
    %\begin{center}
        \includegraphics[scale=0.15]{5fold_equal}
    %\end{center}

    \item creases perpendicular plus one at $45^\circ$ angle \qquad
    %\begin{center}
        \includegraphics[scale=0.15]{5fold_square}
    %\end{center}
\end{enumerate}
\end{prob}
\ul{Discussion}:
\begin{enumerate}[(a)]
    \item In this crease configuration, the unit null cone near the unfolded state 
    is two disjoint circles.
    If we split up this space according to mountain/valley labellings,
    there are $10$ one-dimensional regions and $10$ zero-dimensional regions. 
    \begin{center}
    [INSERT DIAGRAM]
    \end{center}
    
    \item If we split up this space according to mountain/valley labellings,
    there are $4$ one-dimensional regions and $4$ zero-dimensional regions.
    \begin{center}
    [INSERT DIAGRAM]
    \end{center}
\end{enumerate}

\begin{prob}
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?
    
    \item What shapes are the two-dimensional regions? 
    How do they fit together? (i.e. draw out a diagram)
\end{enumerate}


\end{prob}




\end{document}