Hundreds of millions of people ride bicycles every day. Yet only a small fraction are aware of the multitude of beautiful mathematics and physics involved. Describing a bicycle ride requires the mechanics of nonholonomic multi-body systems, control theory, and algebraic geometry.
Drag the front wheel (red point) of an idealized bicycle in the plane. Given its path, the path of the rear wheel is uniquely determined (through a set of coupled nonlinear differential equations). Depending on the length of the wheel base and the curvature of the path of the front wheel, the path of the rear wheel exhibits characteristic cusps.