Mixing Property of the Pedal Mapping

Abstract

In a recent article in the Monthly [2] Peter Lax proved a mixing property of the pedal mapping P which maps the set of angles of a triangle into the angles of the pedal triangle. His result tells us that if we start with almost any triangle, the sequence of successive pedal triangles contains triangles with shapes arbitrarily close to all possible shapes. In this note we represent the pedal mapping as a shift. We could have stopped there and referred to books on ergodic theory for proofs of mixing properties of shift mappings but we thought it would help many readers to give brief derivations of what is needed here, using only the elements of measure theory and probability theory. Lax follows Kingston and Synge [1] in representing a triangle of given shape as a point x in an equilateral triangle E = A PQR. The barycentric coordinates of x are the angles of the triangle in units of r. In this representation the pedal mapping is the result of the following operations. Connect the midpoints of the sides of E. Denote the four congruent triangles into which E is subdivided by EP, EQ, ERand EM where EP is adjacent to P etc. On the interior of a corner triangle the pedal mapping P is the dilatation by a factor 2 which maps the interior of the corner triangle onto the interior of E. On the interior of EM, P is a reflection in the centroid of E combined with a dilatation by 2. The boundaries of the small triangles correspond to right angled triangles and degenerate triangles for which the pedal triangle is degenerate and P is undefined. We want to consider iterates of P. The mapping P is not defined on the boundaries of the four small triangles. Its square is not defined on the preimages of these, which are the boundaries of the subdivisions of EP1,..., EM, etc. We get rid of these loose ends by restricting P to the set E' obtained by removing from E its boundary points and the boundary points of all triangles obtained by successive subdivisions. The removed po,ints form a set of measure 0. Let us choose units so that the measure of E' is 1, so that we can employ the language of probability theory. The mixing property we are going to prove is the following.