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\begin{document}

\begin{center}

\Huge{1 Dimensional Folding}\\
\normalsize{definitions \& theorems from Origametry by Daniel Heath.}

\end{center}

\noindent
While working in a group make sure you:
\begin{packed_itemize}
\item Expect to make mistakes but be sure to reflect/learn from them!
\item Are civil and are aware of your impact on others.
\item Assume and engage with the strongest argument while assuming best intent.
\end{packed_itemize}

\normalsize


\noindent Consider the one dimensional \emph{foldability game}: Begin with a set of two points $S_1=\{A_0, A_1\}$.  We use the convention that the distance between the first two points is one unit and $A_0$ has coordinate 0.  We can create new points to append to the set $S_n$ by using either of the following methods:

\begin{packed_enum}
\item We can use a point $A_i$ as a crease of a fold $\phi$ and apply $\phi$ to another point $A_j$ to create a new point $A_{n+1}=\phi(A_j)$. 
\item We can fold a point $A_i$ onto the top of another such as $A_j$.  This fold $\phi$ will create a new point $A_{n+1}$ at the crease of the fold.  
\end{packed_enum}

\noindent Both of these methods will give us another point that we can add to our set $S_{n+1}=\{A_0, A_1, ....A_n, A_{n+1}\}$.  The set $S_\infty$ is considered to be the set of \emph{foldable points}.  Use these rules for the following:





\begin{enumerate}
\item Find the point with coordinate $\frac{5}{8}$.
\vspace{1in}
\item Find the smallest set of coordinates needed to produce the point with coordinate $\frac{5}{8}$.
\vspace{1in}
\item Mark four points $A$, $B$, $C$, and $D$.  Find a point $E$ so that $C-D-E$ and $DE=AB$.  That is, add the length of $\overline{AB}$ to  $\overline{CD}$. What is the smallest number of folds it takes you?
\vspace{1.5in}


\newpage
\begin{customPost}{C-1}
Let $\lambda >0$ be given.  Given any two distinct points, there is a 1-1 correspondence, called a \emph{ruler}, between all the points on the circle and the real numbers modulo $2\lambda$ that sends one of the two given points to 0 an the other to some number $x>0$.  The number $p$ assigned to a point $P$ is called its \emph{coordinate}.
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\vspace{.3in}
\begin{customDef}{3.1}
If $A$ and $B$ are arbitrary points with coordinates $a$ and $B$ respectively, then the \emph{distance} $AB$ from $A$ to $B$ is defined as: 
$$AB=\begin{cases} 
      \vert a-b \vert & \text{if } \vert a - b \vert \leq \lambda \\
	2\lambda - \vert a-b \vert & \text{if } \vert a - b \vert > \lambda \\

   \end{cases}$$
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\vspace{.3in}

\item Let $\lambda=\pi$. Find the distance between $\frac{2\pi}{3}$ and $-\frac{2\pi}{3}$.  How about between 2 and 10?
\vspace{2in}
\item Recall on the line that $B$ is \emph{between} $A$ and $C$ if $AB+BC=AC$.  Let us define \emph{between} similarly on the circle.  Identify three points on the circle such that no point is between the others.
\vspace{1in}
\end{enumerate}

\end{document}




