The limit of a pedal sequence of triangles

Abstract

Several authors have studied the dynamics of the pedal sequence in which a triangle ABC is replaced by its pedal triangle (whose vertices are the feet of the perpendiculars from each vertex to the opposite side) and this procedure is repeated indefinitely. They found cases where the shape of the triangle repeats periodically and showed that the map is ergodic and modelled by the full shift on four symbols. We show here that the parameter space of shapes of triangles is a tetrahedron and that this has as a double cover (branched over four points) the two-torus where the dynamics is multiplication by 2. The pedal sequence thus inherits its dense set of periodic points and also the mixing property from this toral endomorphism. We find an expression for the limit point of a pedal sequence and show that it depends continuously but not differentiably on the shape of the original triangle.