{\em The material and figures in this section are adapted from the forthcoming textbook ``Masterclasses in Mathematics'' by Ian Stewart and David Wood.}

The wheel has been around for over 6000 years with little change to its design. It is a circle. \\
But can we use other shapes to roll? Might they be better?

The key characteristic we want out of a wheel is that it has a constant {\em diameter}. Namely, whenever it is touching the ground, its top point is always at the same height.

\begin{problem}{Diameter}
    What is the diameter of a circle of radius $r$? \\
    How might you define the diameter of a more obscure shape? \\
    Can a shape have more than one diameter?
\end{problem}
\pagebreak

\definition{Simple Closed Curve} 
A simple closed curve is anything you can draw without picking up your pencil and without crossing over your own line.

\definition{Support Lines}
A support line for a simple closed curve is a line that meets the curve at one or more points, with the extra condition that the curve lies entirely on one side of that line (including points on the line).

\begin{figure}[h]
    \centering
    \includegraphics[width=0.75\linewidth]{img/support lines.png}
    \caption{Some examples and non-examples of support lines.}
\end{figure}

\definition{Diameter}
    A {\em diameter} of a simple closed curve is the distance between two distinct parallel support lines on opposite sides. This diameter can depend on the direction in which the support lines are drawn.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.5\linewidth]{img/diameter.png}
    \caption{Two diameters of a typical curve.}
    
\end{figure}

\pagebreak
\begin{problem}{}
    Briefly explain why pairs of parallel support lines exist. Given a direction, how could you find a pair of parallel support lines systematically?
\end{problem}
\begin{solution}
    Start with a single line in a given direction that doesn't touch your curve. Move the line towards the curve until you find a point of contact. Then continue moving until you find the last point of contact. The first point of contact and the last point of contact give you a pair of parallel support lines.

    Essentially, you are picking the shape up with a pair of chopsticks. This can always be done.
\end{solution}
\vfill
