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We study a simple model of bicycle motion: a segment of fixed length inmulti-dimensional Euclidean space, moving so that the velocity of the rear endis always aligned with the segment. If the front track is prescribed, thetrajectory of the rear wheel is uniquely determined via a certain first orderdifferential equation -- the bicycle equation. The same model, in dimensiontwo, describes another mechanical device, the hatchet planimeter.  Here is a sampler of our results. We express the linearized flow of thebicycle equation in terms of the geometry of the rear track; in dimensionthree, for closed front and rear tracks, this is a version of the Berry phaseformula. We show that in all dimensions a sufficiently long bicycle also servesas a planimeter: it measures, approximately, the area bivector defined by theclosed front track. We prove that the bicycle equation also describes rolling,without slipping and twisting, of hyperbolic space along Euclidean space. Werelate the bicycle problem with two completely integrable systems: the AKNS(Ablowitz, Kaup, Newell and Segur) system and the vortex filament equation. Weshow that "bicycle correspondence" of space curves (front tracks sharing acommon back track) is a special case of a Darboux transformation associatedwith the AKNS system. We show that the filament hierarchy, encoded as a singlegenerating equation, describes a 3-dimensional bike of imaginary length. Weshow that a series of examples of "ambiguous" closed bicycle curves (fronttracks admitting self bicycle correspondence), found recently F. Wegner, arebuckled rings, or solitons of the planar filament equation. As a case study, wegive a detailed analysis of such curves, arising from bicycle correspondencewith multiply traversed circles.

Tire tracks and integrable curve evolution