The Symbolic Dynamics of the Sequence of Pedal Triangles

Abstract

It is standard in plane geometry to construct the three medians, angle bisectors, or altitudes of a given triangle T. The three lines of any set intersect the opposite sides (or their extensions) of the triangle in three points (the feet) that can be taken as the vertices of a new triangle T'. The process can be iterated. Of course the successive triangles become smaller, but we concentrate only on their angles. For example, the triangle formed from the feet of the medians is similar to the original triangle; the angles are unchanged. The successive triangles formed from the angle bisectors limit to an equi-angular triangle. On the other hand, the angles of the successive triangles formed from the feet of the altitudes behave in a much more complicated manner. In some sense, every type of behavior is possible. For any r, there are triangles whose angles repeat after r iterations (for example, a triangle with angles 360, 720, 720 is similar to its second iterate; a triangle with angles 120, 360, 1320 is similar to its fourth iterate). For any r and k, there are triangles whose angles repeat every r iterations after an initial delay of k iterations. For each k, there are triangles that after k iterations are equiangular (for example, 600, 105?, 15? with k = 2). There are lots of triangles whose angles never repeat (uncountably many). Indeed, there are triangles whose angles come arbitrarily close to any given triangle. There are triangles that become almost equiangular and stay that way for, say 1,000,000 iterations, but that then do something completely different. Hobson [3] called the triangle T' formed from the feet of the altitudes of a triangle T the pedal triangle. Recently Kingston and Synge [4] revisited and corrected Hobson's work and, for example, determined a criterion for some pedal iterate of T to have the same angles as T. The purpose of the present article is to revisit the issue again from a different mathematical point of view, which makes it routine to understand the behavior of the angles of successive pedal triangles. Let us consider the question of triangles whose angles repeat after r pedal iterations. Determining criteria directly on the angles leads to rather technical conditions [4] (and in fact is where Hobson made mistakes). A recurring theme in mathematics is to relabel objects so that desirable properties are more evident, and that is our present theme. For each triangle T, we assign a label E(T) that makes the behavior under the iterated pedal map obvious. Equally important, there is a straightforward way of determining the angles of the triangle from its label. More precisely, we consider all infinite sequences a1a2a3... of four symbols, say each ai equals 0, 1, 2, or 3. Every triangle T is labelled with one such sequence E(T), and two triangles are labelled with the same sequence if and only if they are