Gil Bor, Bicycle Tracks, their Monodromy Invariants and Geodesics

At first sight, the pair of front and back wheel tracks left by a passing bike on a sandy or muddy terrain seems like a random pair of curves. This is not the case. For example, one can usually distinguish between the front and back wheel tracks, and even the direction at which these were traversed, based solely on their shapes. You can try it for the following pair of paths.

Another example: If the front wheel traverses a small enough closed path (compared to the bike size), then, typically, the back track does not close up, by an amount approximated by the area enclosed by the front track and the bicycle length; this fact was utilized to build a simple area measuring mechanical device, now obsolete, called the Hatchet planimeter.

In recent years the subject has attracted attention due to newly discovered relations with the theory of completely integrable systems (the filament flow), sub-Riemannian geometry and elasticity theory. I will try to describe some of these developments and open questions.