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\documentclass{article}
\usepackage{graphicx} % Required for inserting images
\usepackage{enumerate}
\title{Holonomy in Colors}
\author{Stephanie Atherton}
\date{June 2024}

\begin{document}

\maketitle

Stephanie was at a special math program this past summer where she rolled shapes on other shapes - or studied \emph{holonomy} (in fancy math terminology). For example, consider the colored tetrahedron below which we will be rolling on all our surfaces/shapes throughout this column. Note that the regular tetrahedron has \\4 faces,\\
6 edges,\\ 
and 4 vertices,\\
with all its polygonal faces and angles of vertices being identical.  

\begin{center}
\includegraphics[scale = .75]{pics/nws2.png}
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Always starting with the colored tetrahedron face-down and aligned with a face on a triangulated surface, you will be rolling our colored tetrahedron in full "lassos" over edges and around a single vertex.  See the below illustration of a lasso within the triangulated hexagon of the triangulated surface:
\begin{center}
\includegraphics[]{pics/nws1.png} 
\end{center}
 A single lasso path starts and ends at the same face but can be rolled either clockwise or counterclockwise.
Pretend every time you complete a lasso on a face, you are "stamping" the color of the tetrahedron of that face on the triangle of the given surface. \\

To warm up, let's imagine a triangulated plane like the image above that goes on forever in every direction - the plane is \emph{infinite}. Then there is no end to the plane - so we can roll our tetrahedron all around in any direction as many times as we want!
\begin{enumerate}
    \item Starting with the magenta face face-down, what possible colors can you produce as you roll in lassos on the infinite surface? And let us consider now the \textit{orientation} of the tetrahedron - are all vertices of the tetrahedron in the same ending position as starting position when you complete a lasso?  Or is the tetrahedron in a different orientation from when you began rolling?  
      \item We've shown you the 3D version, now draw what a tetrahedron might look like if you were to unfold it as a flat surface, producing what is known as the \textit{geometry net} of the tetrahedron.Find the two possible geometry nets for the tetrahedron.
    \item Rolling the colored tetrahedron in lassos on the geometry net of a tetrahedron and always starting with indigo face face-down, what possible colors can you produce?
    \item How many total lasso paths did you find by rolling the tetrahedron on its net?  How many edges did you roll over each for each lasso?  Don't be afraid to draw it out and show your work!
\begin{center}
\includegraphics[scale = .25]{pics/Octahedron.svg.png}    
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\item Now consider and find all possible geometry nets for the regular octahedron (pictured above).  We will be rolling our tetrahedron on these nets.
\item  How many total \textit{permutations} (considering the orientation of each of the tetrahedron's faces) can you find for the colored tetrahedron rolled on the octahedron?  Please list the possible permutations. (Hint: Consider numbering the vertices of your colored tetrahedron.)
\end{enumerate}
Keep rolling and figure out the answers!
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