Motivation:
Puzzle: Draw $\odot$ on paper without lifting your pen.
Solution: Start by putting your pen in the centre of the paper to create the dot. Then fold the paper such that the edge of it touches your pen tip. Draw a line outward from the dot but on the other side of the paper. With the paper up against the original side until necessary, continue to draw the outer circle.
Could we generalise this solution using origami and calculus (e.g., continuity)?
The Question:
Suppose we have a square piece of paper and a pen. Can we make precise what it means for us to draw such shapes/curves "continuously"?
Essentially, I mean: how do we describe rigorously the simple process of drawing on folded paper without lifting the pen, allowing for unfolding?
Thoughts:
My guess is that we could use origami and calculus, though I don't know much about mathematical origami. My calculus is rusty. Piecewise functions might be useful.
Maybe the domain of our functions could be $[0,1]$ for simplicity.
Fun thought: I think is continuous. (For those who can't see, it's a smiley face $\ddot\smile$ inside a circle $\bigcirc$.)