A ring, a torus, a genus one thing
Is it possible to draw a straight line on a tyre?
On the left is an image of practice drawings of straight lines on the surface of a polystyrene torus. Euclid defined a straight line as a line that lies evenly with itself and that is the technique I'm working on: drawing the next little bit of the line 'evenly' with itself. On the right, is a wiggledy line that follows the bumps on a notsmooth torus.

Different structures on surfaces constrain the sorts of curves that can be 'drawn'. When the surface is not smooth, as the left hand torus below, is is not possible to draw a 'straight line'. On the torus on the right, two geodesics are represented by stretched plastic strip clinging to the surface. Maryam Mirzakhani's work 'counted' curves in terms of other shapes (so, for example, instead of saying there are 108 possible curves, one might say that there are a sphere's worth).
to be continued ...